University  of  California  •  Berkeley 

The  Theodore  P,  Hill  Collection 

of 

Early  American  Mathematics  Books 


KEY 


TO    THE 


PROGRESSIVE 


PEACTICAL  ARITHMETIC. 


INCLUDINa 


ANALYSES  OF  THE  MISCELLANEOUS  EXAMPLES 


PROGRESSIVE  INTELLECTUAL  ARITHMETIC. 


FOR    TEACHERS    ONLY. 


IVISON,  BLAKEMAN,  TAYLOR  &  CO.. 

NEW   YORK   AND    CHICAGO. 

1877. 


ROBINSON^S 

Mathematical   Series. 

Graded  to  tlie  wants  of  Primary,  Intermediate,  Grammar, 
Normal,  and  High  Schools,  Academies,  and  Colleges. 


Progressire  Table  Boob. 
Progressive  Primary  Aritliraetic. 
ProgressiYO  Intellectual  Arithmetic, 
Rudiments  of  Written  Arithmetic. 

JUNIOR-CLiSS  ARITHMETIC,  Oral  and  Written.    NEW. 
Progressive  Practical  Aritlimetic. 
Key  to  Practical  Arithmetic. 
Progressive  Higher  Arithmetic. 
Key  to  Higher  Arithmetic. 
Nevr  Elementary  Algebra. 
Key  to  New  Elementary  Algebra. 
New  University  Algebra. 
Key  to  New  University  Algebra. 
New  Geometry  and  Trigonometry,    In  one  vol. 
Geometry,  Plane  and  Solid.    In  separate  vol. 
Trigonometry,  Plane  and  Spherical.    In  separate  vol. 
New  Analytical  Geometry  and  Conic  Sections. 
New  Surveying  and  Navigation. 
New  Differential  and  Integral  Calculus. 
University  Astronomy— Descriptive  and  Physical. 

Key  to  Geometry  and  Trigonometry,  Analytical  Geometry  and  Conie 
Sections,  Surveying  and  Navigation. 


Entered,  according  to  Act  of  Congress,  in  the  year  1860,  by 

HORATIO    N.    ROBINSON,    LL.D., 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Northern 
District  of  New  York. 


P  R  E  I^  1  0  1 . 


A  Key  to  any  Matliematical  work  is  not  intended  to  sa- 
persede  labor  and  study,  but  to  give  direction  to  the  latter  and 
make  it  more  effective  and  useful. 

In  many  examples  and  problems  tbe  same  results  may  be 
obtained  by  different  processes,  but  the  shortest  and  most 
simple  method  is  to  be  desired ;  hence  the  object  of  a  Key 
should  be  to  give  not  results  only,  but  the  explanation  of 
methods,  and  a  full  analysis  of  such  questions  as  contain  a 
peculiar  application  of  principles  involved. 

It  is  supposed,  of  course,  that  every  teacher  is  fully  com- 
petent to  solve  all  the  questions,  but  with  the  multiplicity 
of  duties  ordinarily  put  upon  the  teacher,  time  cannot  always 
be  had  to  answer  or  solve  all  the  questions  presented  by  the 
pupil.  Therefore  the  Key  is  intended  to  lessen  the  labor 
and  save  the  time  of  the  teacher  by  presenting  the  shortest 
solution,  and  the  best  form  of  analysis  as  a  standard  to  which 
the  pupil  should  be  required  to  conform. 


IV  PREFACE. 

In  compliance  with  the  wishes  of  many  teachers,  brief 
analyses  of  the  Miscellaneous  Examples  in  the  Intellectual 
Arithmetic  have  been  added  to  the  latter  part  of  this  work. 

Much  labor  has  been  bestowed  upon  the  present  work 
to  give  a  full,  complete,  and  logical  analysis  of  all  difficult 
rxamples^  and  of  g'jch  questions  as  contain  the  application 
of  a  new  principle,  The  arrangemeut  i&  such  as  to  be 
easily  understood. 


KEY. 


NOTATION. 


ROMAN     NOTATION  . 


(17,  page  9.) 


Ex. 
Ex. 
Ex. 
Ex. 


1.  Ans   XL 

S,  Ans,  XXV. 

5.  Ans.  XLVm. 

Y.  Ans.  CLIX. 

9.  Ans.  MDXXXYIII. 


Ex.  2.  Ans.  XV. 
Ex.  4.  Ans.  XXXIX. 
Ex.  6.  Ans.  LXXVII. 
Ex.  8.  Ans.  DXCIV. 
Ex.  10.  Ans.  MDCCCCX, 


I 


Ex. 
Ex. 
Ex. 


1.  Ans.  125. 
4.  Ans,  900. 
1.  Ans.  505. 


ARABIC    NOTATION. 

(26,  page  12.) 
Ex.  2.  Ans.   483. 


Ex.5. 
Ex.  8. 


Ans.   290. 
w4?i5,  557. 


Ex.3.  Ans.   716. 
Ex.  6.  ^/i5.  809. 


(28,  page  13.) 

Ex.  2.  ^W5.  5160. 
Ex.  5.  ^715.  2090. 
Ex.  8.  Ans.   9427. 


Ex.  3.  Ans.  3741. 
Ex.  6.  ^715.  7009. 
Ex.  9.  Ans.   4035. 


Ex.  1.  Ans.  1200. 
Ex.  4.  ^715.  8056. 
Ex.  7.  Ans.  1001. 
Ex.  10.  ^»5.  1904. 

Ex.  11.  Ans.  Seventy-six;  one  hundred  twenty-eight ;  four 
hundred  five ;  nine  hundred  ten  ;  one  hundred  sixteen ;  three 
thousand  four  hundred  sixteen  ;  one  thousand  twenty-five. 

Ex.  12.  Ans.  Two  thousand  one  hundred;  five  thousand 
forty-seven ;  seven  thousand  nine  ;  four  thousand  six  hundred 


6  SIMPLE   NUMBERS. 

seventy;  three  thousand  nine  hundred  ninety  seven;  one 
thousand  one. 

(29,  page  14.) 

Ex.  1.  Ans.  20000.  Ex.  2.  Ans.  47000.  Ex.  3.  Ajis.  18100. 
Ex.  4.  Ans.  12350.  Ex.  5.  Ans.  39522.  Ex.  6.  Ans.  15206. 
Ex.  7»  Ans.  11024.  Ex.  8.  Ans.  40010.  Ex.  9.  Ans.  60600. 
Ex.  10.  Ans.  220000.  Ex.  11.  Ans.  156000. 

Ex.  12.  Ans.  840300.  Ex.  13.  Ans.  501964. 

Ex.  14.  Ans.  100100.  Ex.  15.  Ans.  313313. 

Ex.  16.  Ans.  718004.  Ex.  17.  Ans.  100010. 

Ex.  18.  Ans.  Five  thousand  six;  twelve  thousand  three 
hundred  four  ;  ninety-six  thousand  seventy-one  ;  five  thousand 
four  hundred  seventy  ;  two  hundred  three  thousand  four  hun- 
dred ten. 

Ex.  19.  Ans.  Thirty-six  thousand  seven  hundred  forty-one  ; 
four  hundred  thousand  five  hundred  sixty  ;  thirteen  thousand 
sixty-one  ;  forty -nine  thousand ;  one  hundred  thousand  ten. 

Ex.  20.  Ans.  Two  hundred  thousand  two  hundred  ;  seventy 
five  thousand  six  hundred  twenty  ;  ninety  thousand  four  hun- 
dred two ;  two  hundred  eighteen  thousand  ninety-four ;  one 
hundred  thousand  one  hundred  one. 

(31 5  page  16.) 

Ex  1.  Ans.  140.  Ex.  2.  Ans.  30201.  Ex.  3.  Ans.  8050. 
Ex  4.  Ans.   2900417.  Ex.  5.  Ans.   300040. 

Ex.  6.  Ans.  96037009.  Ex.  7.  Ans.  4064200150, 
Ex.  8.  Ans.   846009350208. 

(34,  p.  19.) 

Ex.  1.  Ans.   436.  Ex.  2.  Ans.   7164. 

Ex.  3.  Ans.   26026.  Ex.  4.  Ans.   14280. 

Ex„  5.  Ans.   176000.  Ex.  6.  Ans.   450039. 

Ex.  7.  Ans,   95000000.  Ex.  8.  Ans.   433816149. 


NOTATION   AND    NUMERATION.  7 

Ex.9.   Ans,  900090.  Ex.  10.  Ans.  10011010. 

Ex.  11.  Ans.  61005000000.      Ex.  12.  Ans,  5080009000001. 

Ex.  13.  Ans,  Eight  thousand  two  hundred  forty. 

Ex.  14.  Ans,  Four  hundred  thousand  nine  hundred. 

Ex.  15.  Ans.  Three  hundred  eight. 

Ex.  16.  Ans,  Sixty  thousand  seven  hundred  twenty, 

Ex.  17.  Ans,  One  thousand  ton. 

Ex.  18.  Ans,  Fifty-seven  million  four  hundred  sixty-eight 
thousand  one  hundred  thirty-nine. 

Ex.  19.  Ans,  Five  thousand  six  hundred  twenty- eight 

Ex.  20.  Ans,  Eight  hundred  fifty  million  twenty-six  thou- 
eand  eight  hundred. 

Ex.  21.  Ans,  Three  hundred  seventy  thousand  five. 

Ex.  22.  A71S,  Nine  billion  four  hundred  million  seven  hun- 
dred six  thousand  three  hundred  forty-two. 

Ex.  23.  Ans,  Thirty-eight  million  four  hundred  twenty-nine 
thousand  five  hundred  twenty-six. 

Ex.  24.  Ans,  Seventy-four  billion  two  hundred  sixty-eight 
million  one  hundred  thirteen  thousand  seven  hundred  fifty- 
nine. 

Ex.  25.  Ans,  7000036. 

Ex.  26.  Ans,  563004. 

Ex.27.  Ans,  1096000. 

Ex.  28.  Ans,  Nine  billion  four  million  eighty-two  thousand 
five  hundred  one. 

Ex.  29.  Ans,  Two  trillion  five  hundred  eighty-four  billion 
five  hundred  three  million  nine  hundred  sixty-two  thousand 
forty-seven, 

Ex.  30.  Ans.  3064159, 

Ex,  31.  A71S.  Two  of  the  sixth  order,  9  of  the  fifth,  6  of  the 
third,  4  of  the  second,  and  8  of  the  first. 

Ex.  32.  Ans.  One  of  the  seventh  order,  3  of  the  fifth  order, 
7  of  the  fourth  order,  and  5  of  the  second  order. 


Ex.  3 
Ex.6. 


Ans.  6i>8. 
Ans.  898. 


SIMPLE   NUMBElia 


ADDITION. 

(40,  page  21.) 

Ex.  4.   Ans.  967. 


(43,  page  24.) 

Ex.    7.   ^7i5.  1807.  Ex.    8.  Ans.  27246. 

Ex.    9.    Ans.  4945.  Ex.  10.  ^7Z5.  78313. 

Ex.  19.   Ans.  145.  Ex.  22.  Ans.  69585. 

Ex.  23.    Ans.  566.  Ex.  24.  Ans.  3746. 

Ex.  27.   ^^^5.  4619.  Ex.  28.  Ans.  4915. 

Ex.  29.    Ans.  4320.  Ex.  30.  Ans.  4623. 

Ex.  31.   Ans.  3871.  Ex.  35.  ^?Z5.  101500. 

Ex.37,   ^ns.  50000000.  Ex.40,  ^w*.  1194. 

Ex.  44.   Ans.  2773820.  Ex.  45.  Ans.  4403241. 


Ex.  e. 

Ex.  8. 
Ex.  19. 
Ex.  24. 
Ex.  26. 


SUBTKACTION. 
(49,  page  30.) 


Ans.  353. 
Ans.  205. 
Ans.  123. 
Ans.  4202. 
Ans.  16348755. 


Ex.  7. 
Ex.  9. 
Ex.  22. 
Ex.  25. 

Ex.  27. 


Ans.  210. 
Ans.  320. 
Ans.  2113. 
Ans.  11425, 
Ans.  4014580. 


(51,  page  32.) 


Ex.    3.   Ans.  721. 

Ex.    4.    Ans.  561. 

Ex.    5.   ^ws.  3769. 

Ex.    6.    Ans.  269, 

SUBTRACTION. 

Ex. 

7. 

Arts.  4509. 

Ex.  8. 

Ans.  3449. 

Ex. 

9 

Arts.  1288. 

Ex.  10. 

Ans.  30616. 

Ex. 

11. 

Ans.  21078. 

Ex.  12. 

Ans.  142. 

Ex. 

13. 

Ans.  762301. 

Ex.  19. 

Ans.  224130. 

Ex 

20. 

Ans.  220874. 

Ex.  25. 

Ans.  181972. 

Ex. 

31. 

Ans.  529509693. 

Ex.  34. 

Ans.  1902001 

EXAMPLES    COMBINING    ADDITION    AND    SUBTRACTION, 


(52,  page  33.) 

Ex.  1.     2500+    175  =  2675 

6200—2675  =  2525  dollars,  Ans. 

Ex.  2.     235  +  275  +  325  +  280  =  1115; 
1300  —  1115  =  185  miles,  Ans. 

Ex.  3.     4234  +  1700  +  962  +  49  =  6945, 
87,^^-6945  =  1807  dollars,  Ans. 

Ex.  4.     47(^5  +  750=5515; 

5515  —  384  =  5131  dollars,  Ans. 

Ex.  6.     1224  +  1500  +  1805  =  4529; 

7520  —  4529  =  2991  barrels,  Ans. 

Ex.  6.       450  +  175=  625,  B's; 

450  +  625  =  1075,  A'sandB's; 
1075  —  114=  961,  C's  sheep,  Ans. 

Ex.  7.     1575  —  807  =  768,  bushels  of  wheat, 
900  —  391  =  509,       "        "  corn, 

Ex.  8.     2324  +  1570  +  450  +  175=4519; 
6784—4519  =  2265  miles,  Ans. 


at,   ) 


Ex.  9.  7375,  first  paid; 

7375+   7375  =  14750,  second  paid; 
7375  +  14750  =  22125; 
36680  —  22125  =  13555,  dollars,  Ans. 


Ann, 


10 


SIMPLE   NUMBERS. 


Ex.  10.     '750-{-3'79  +  450  =  1579; 

1579  — 1000=579,  dollars,  Ans. 

Ex.  11.  6709  +  3000=9709; 

9709—4379  =  5330  dollars,  Ans, 

Ex.12.     10026402+    9526666  =  19553068,  total ; 
8786968+   8525565  =  17312533,  native; 
19553068  —  17312533=   2240535,  foreign,  Ans. 


MULTIPLICATION. 


(61,  page  38.) 


Ex. 

5 

Ans.  247368. 

Ex.  6. 

Ans. 

648998. 

Ex. 

7. 

Ans.  224744. 

(64, 

Ex.  8. 
page  41.) 

Ans. 

416223. 

Ex. 

5. 

Ans.  2508544. 

Ex.  6. 

Ans. 

15731848, 

Ex. 

7. 

Ans.  16173942. 

Ex.  9. 

Ans. 

78798. 

Ex. 

13. 

Ans.  937456. 

CONTRACTIONS. 


Ex. 

2. 

Ex. 

3. 

Ex. 

4. 

Ex.  5. 


(675  page  43.) 

3472x6  =  20832;  20832x8  =  166656,  Ans. 
14761  X  8  =  118088  ;  118088  x  8=944704,  Ans. 
87034  X  3  =  261102  ;  261102  x  3  =  783306  ; 

783306x9  =  7049754,  Ans. 
47326  X  6  =  283956  ;  283956  x  5  =  1419780  • 

1419780x4  =  5679120,  Aiis. 


MULTIPLICATION,  II 

Ex.    6.  60315x8X3x4  =  5790240,  Ans, 

Ex.    7.  291042x5x5x5  =  36380250,  Ans. 

Ex.    8.  430  X  7  X  8  =  24416  miles,  Ans. 

Ex.     9.  124  X  6  X  3  X  4  =  8928  dollai-s,  Ans. 

Ex.13.  5280x7x3x4  =  443520  feet,  Ans. 

Ex.  11.  120x5x5x5  =  15000  dollars,  Ans. 

{61>5  page  44.) 
Ex.  3.   Ans.  13071000.  Ex.  4.   Ans.  890170000. 

(70,  page  45.) 
Ex.  12.     296 

3000 


888000  dollars,  Ans. 

EXAMPLES    COMBINING    ADDITION,    SUBTRACTION,    AHD 
MULTIPLICATION. 

Ex.    1.     4x45  =  180;    13x9  =  117; 

180  +  117=297  dollars,   Ans. 
Ex.    2.     31x6  =  186;    39x6  =  234; 

234  —  186  =  48  dollars,    Ans, 

Ex.    3.     288x9  =  2592; 

2592  —  1875  =  717  dollars,   Ans. 

Ex.    4.     240  +  125  +  75  +  50  =  490; 

500  — 490  =  10  dollars,   Ans, 
Ex.    5.     184x2=368;  67x4  =  268; 

368  —  268  =  100  dollars,  Aiis 

Ex.  6.  36  X  320  =  11520,  A  received  ; 
48  X  244  =  11712,  B  received  ; 
11712  —  11520  =  192  dollars,    Ans. 

Ex,  7  35  +  29  =  64  miles,  in  one  day  ; 
64  X  16  =  1024  miles,  ^^. 


12  SIMPLE   NUMBERS. 

E%.    8.     14  X  26  X  43  =  15652  yards,  Am* 
Ex.    9.     4  X  365=:  1400,  yearly  expenses  ; 
3700—1460  =  2240  dollars,  Ans, 

Ex.10.  2475,  first; 

2475—  840  =  1635,  second; 
2475  +  1635  =  4110,  third; 

8220  dollars,  Ang. 

Ex.  II.     336  — (28  X  10)  =  56  miles.  Am. 

Ex.12.     23  X    14=  ^22,  cost  of  cows; 
96  X      7=  672,   "     "   horses; 
57  X    34  =  1938,    "     "    oxen; 
2x300=  600,   "     "   sheep; 

3532,    "     "   whole. 
3842  —  3532  =  310  dollars,   Ans. 

Ex.  13.     36X164  =  5904 
Sx  850  =  2550 


8454  dollars,  Ans. 

Ex.  14.     14760— (1575  x  5)=6885  dollars,  An9» 

Ex.  15.     936  X  9=8424,  cost ;  480  x  10  =  4800 

456  X    8=3648 


Flour  sells  for,       8448. 
8448  —  8424  =  24  dollars,  Ans. 


DIVISION. 

(77,    page  50.) 

Ex.    2.   Ans,  16358.  Ex.    3.   Ans,  17827. 

Ex.    4.    Ans.  29822.  Ex.    5.    Ans.  672705. 

Ex.    6.    Ans.  182797.  Ex.    7.    Ans.  829838. 


DIVISION.  13 

Ex.  13.  Ans,  1048795f.     Ex.  14.  Ans,  635926f . 

Ex.  15.  Ans.  2379839I-.     Ex.  16.  Ans.  9355Y51f 

Ex.  17.  Ans,  245640}-!.     Ex.  18.  Ans.  70141321. 

Ex.  19.  47645-^5  =  9529  dollars,  Ans.      • 

Ex.  20.  17675-^7:=:2525  weeks,    Ans. 

Ex.  21.  6756^6  =  1126  barrels,    Ans. 

Ex.  22.  46216464^12:^3851372  dozen,    Ans. 

Ex.23.  347560-^5  =  69512  barrels,    Ans. 

Ex.24.  3240622-7-11  =  294602  acres,    Ans. 

Ex.  25.  38470-T-5=7694  dollars,    Ans. 

(80,  page  54.) 

Ex.    5.  Ans.  212//4-.  Ex.  14.    Ans.  1489f|. 

Ex.  15.  Ans.  121522%.  Ex.  16.   Ans.  508301yV2- 

Ex.  17.  Ans.  12109001$ Jf      Ex.  18.   Ans.  9974^9-. 

Ex.  19.  Ans.  1343if  Jf.  Ex.  20.    Ans.  5473|fff 

Ex.  21.  Ans.  7500yVTTV 

Ex.27.  1850400-^18504  =  100  dollars,   Ans. 

Ex.  28.  72320060-^10735  =  6736yVT3J  <iollars,   Ans. 

Ex.29.  942321-^213  =  44243^?  3- volumes,   Ans. 

Ex.  30.  5937120^22320  =  266  dollars,    Ans. 


CONTRACTIONS. 


(81 5  page  56.) 

Ex.  2.  (3690-T-3)-4-5  =  246,    Ans. 

Ex.3.  (3528^4)-^6  =  147,    Aiis. 

Ex.4.  (7280-^5)^7  =  208,    Ans. 

Ex.  5  (6228-^6)-^6  =  l73,    Ans. 

Ex.6.  (33642^3)-^9  =  124d,    Ans. 

Ex,  7.  (153160-^7)^8  =  2735,   Ans. 

Ex.  8.  [(15625-^5)-^5]^5  =  125,    An9. 


14  SIMPLE    NUMBERS 

(82,  page  57.) 
Ex.  2.     6)34712 

7)5785 2 

826 ---3x6  =  18^ 

20,  Ans. 
Ex.  3.     8)401376 
8)50172 

6271 ---4x8=32,  Ans. 
Ex.  4.     3)139074 
4)46358 

6)11589 2x3=   6 

1931 ---3x4x3  =  36^ 

42,  Ans^ 
Ex.  5.  3)9078126 
5)3026042 
6)605208  -  -  -  2  X  3=6,  Ans. 
100868 
Ex.  6.  4)18730627 

5)4682656 8 

6)936531 1x4=  4 

156088 ---3x5x4  =  60 

67,  Am. 
Ex.  *i      2)7360479 

6)368C239 1 

8)613373 1  x2=  2 

76671 ---5x6x2  =  60^ 

63,  Ana. 


SIMPLE   NUMBERS.  15 

Ex.  8.     2)24726300 

6)12363150 

7)2472630 

353232  -"GxbX  2  =  60,  Am 
Ex.  9.     7)5610207 


2)801458 1 

6)400729 

66788--- 1x2x7  =  14 

15,  Ans, 

(83j  page  58.) 

Ex.  2.    Ans.  476.  Ex.    3.  Ans.  2Q20j\\. 

Ex.  4.    Ans.  1306y3-2-i7.  Ex.    5.  Ans.  976yf  JJ^. 

Ex.  6.   Ans.  2037lyVoVoV- 

(8 5,  page  59.) 

Ex.  6.   Ans.  14556yVoVoV-  Ex.    7.   ^ns.  106099VoVo- 

Ex.  8.   Ans.  114304^1^15.        Ex.  10.   Ans.  Q84J^^\%\\%\% 
Ex.  11.    24898^50  =  497ff  hours,  Ans. 
Ex.  12.   350000-r-14000=25  dollars,  Ans. 


EXAMPLES    IN   THE    PRECEDING   RULES. 

(Page  60.) 

Ex.    1.     1732  +  67  =  1799,   Ans. 

Ex.    2.     1095-^365=3  dollars,  Ans. 

Ex.    3.     141+47  =  188  dollars,  ^ns. 

Ex.    4.     500  +  17  +  98  +  121  =  736  acres  owned; 

736  —  325=411  acres,  Ans. 
Ex.    6.     2300  — 625  =  1675  dollars,  ^719. 


16  SIMPLE    NUMBERS. 

Ex.    6.     60  —  45  =  15  dollars,  saved  in  one  month; 

900-^15r^60  months,  Ans. 
Ex.    7.     87  X  9  =  783  days,  Ans. 
Ex.    8.  4  first  number ; 

4x8=  32  second; 
32x9  =  288  thh'd; 

324,  Ans. 
Ex.    9.     2x2x7  =  28; 

364^28  =  13,  Ans. 
Ex.  10.     78  +  104  =  182; 

182  X  2  =  364  acres,  Ans. 
Ex.11.     90-1-30  +  12-1-5  +  7  =  144; 

144  X  27  =  3888  dollars,  Ans. 
Ex.  12.     (2250  X  4)^3  =  3000  dollars,  Ans. 
Ex.  13.     35  +  40=75  miles  in  one  day; 

75  X  6  =  450  miles,  Ans. 
Ex.  14.     40—35  =  5  miles  in  one  day; 

5  X  6  =  30  miles,  Ans. 

45  —  19=26  years,  Ans. 

1000000000-7-25000=40000  acres,  Ans. 

384  +  1562  +  25  +  946  =  2917 ; 2917— rr23~l94 
194-f-97=2;  and  2  x  142  =  284,  JLns. 

5280^3  =  1760  steps,  Ans. 
2375  +  340        =2715  dollars,  cost; 
867  +  (235x8)  =  2747       "       sold  foi 
2747  —  2715=32  dollars  gain,  Ans. 
Ex.20.     4500  — 1350  =  3150  to  gain; 

800—   450  =  350  yearly  savings; 
3150^350  =  9  years,  Ans. 
Ex.  21.     1600  X  75  =  120000  ; 

120000-^40  =  3000  bushels,  Ans. 
Ex.  22.     325  X  50  x  2  =  32500  dollars,  Ans. 


Ex. 

15. 

Ex. 

16. 

Ex. 

17. 

Ex. 

18. 

Ex. 

19. 

SIMPLE   NUMBERS.  17 

Ex.  23.     225  —  75  =  150; 

150  X  52  =  7800  cents,  Ans. 
Ex.  24.     31383450^4050  =  7749,  Ans. 
Ex.  25.     31647000-^700  =  45210  dollars,  Ans. 

Ex.  26.-  Reversing  fourth  operation,  100—40=  00; 
Reversing  third  operation,  60  x  5  =  300  ; 
Reversing  second  operation,  300-r-3  =  100; 
Reversing  first  operation,       100  x  7  =  700,  Ans, 

Ex.  27.  (54  X  17)4-27  =  34  cows,  Ans. 

Ex.  28.  56  — (2  X  2  6)  =  4  dollars,  Ans. 

Ex.  29.  98  X  7  =  686  days,  Ans. 

Ex.  30.  5301212-M1137=476  dollars,  Ans. 

Ex.  31.     60  —  39  =  21  gallons,  gained  hourly  ; 
840^21  =  40  hours,  Ans. 

Ex.  32.     4500  X  24  =  108000,  Ans. 

Ex.  33.     1900—1492=408  years,  Ans. 

Ex.34.     Maine,  31766; 

New  Hampshire,     9280 ; 
Vermont,  10212; 

Massachusetts,        7800 ; 
Rhode  Island,  1306 ; 

Connecticut,  4674 ; 

65038—47000  =  18038,  Ans, 

Ex.  36.     25000^8  =  3125  pounds  in  the  thread  ; 
3125  +  235  =  3360  pounds,  Ans. 

Ex.  37.     8546  +  342  =  8888 ; 

8888-^4  =  2222  dollars,  Ans. 

Ex.  38.     245x37  =  9065  ; 

9065  +  230  =  9295,  Ans. 

Ex.  39.     6190048-^72084  =  72,  Ans. 


18  PRIME   NUMBERS. 

Ex.  40.     109  X  73  =  7957,  greater  number; 
28  X  17^:476,  difference; 
7957-476  =  7481  less,  Jns. 

Ex  41.     360  — 114  =  246,  greater; 
246x114  =  28044,  Ans. 

Ex:  42.     2568754  —  2473248  =  95506,  A7i8. 

Ex.  43.     Wheat,  35  x  2  =  70  dollars ; 
Wood,  18x3  =  54        " 

124        " 
Cloth,      9x4  =  36        " 

88  dollars,  Ans. 

Ex.  44.     684  —  375  =  309  yearly  savings  ; 
309  X  5  =  1545  dollars,  Ans. 

Ex.  45.     58  +  10-1-5  +  28  +  3  =  104,  cost  of  one  barrel; 
125  —  104  =  21  cents,  Ans, 

Ex.  46.     286000-6000  =  280000  ; 

280000-^14  =  20000  dollars,  Ans. 

Ex.47.     256x25  =  6400;  6400—625=6775; 
5775-v-35  =  165,  Ans. 

Ex.  48.     189-=-(4  +  5)  =  21  hours,  Ans. 


PRIME  NinVIBERS. 
(91 5  page  68.) 


Ex  2.  Ans,  2,  2,  3,  5,  19.  Ex.  3.  Ans.  3,  3,  5,  5,  7,  II' 

Ex.  4.  Ans,  11,  13, 17.  Ex.  5.  Ans,  19,  23,  29. 

Ex.  6.  Ans,  2,  3,  5,  7,  11  Ex.  7.  Ans,  3,  3,  5,  7,  7. 

Ex.  8.  Ans,  11,  31,  41. 


PRIME    NUMBERS. 


19 


Ex.  2.     24=  i 


Ex.  4.     40  r=  ^ 


(92,  page  69.) 

< 

^^'^^           Ex.3.  12o=r 
3x8 

(5x25 
( 5x6x6 

4x6 

2x3x4 

2x2x6 

2x2x2,>.  3 

2x20 

4x10 

5x8 

2x2x10 

2x4x5 

2x2x2x5 

^2x36 

3x24 

4x18 

6x12 

8x9 

2x2x18   - 

2x3x12 

12=  < 

2x4x9 

3x4x6 

2x6x6 

3x3x8 

2x2x2x9 

2x2x3x6 

2x3x3x4 

*>x2x2x3xa 

20 


PROPERTIES  OF   NUMBERS. 


CANCELLATION. 


Ex. 

3. 

u 

(95,  pa 

n 

u 

ige  72.) 
Ex.  4. 

1$ 

n  8 
11 

^0 

14,  Ans. 

33,  Ans. 

Ex. 

5. 

1  $4 

X$ 

H 

7 

£$  16 

.%$  4 
$ 
64 

Ex.  6. 

30 

$ 

n 
n 

W0 

13 

t 

3 

13 

9|,  Jns. 

41,  -4ms. 

Ex. 

7. 

8  0 
8 

00  3 
9 

X$  3 

81 

Ex,    8. 

tn 

1$ 
u 

200 
?0  2 
$0 

2,  Ana. 

lOi,  Ans. 

Ex. 

9. 

0 

n 

00  2 

Ex.  10. 

4 

0 

$ 
t 
0 

i?2 

0 

*< 

S,Ans. 

^^2 

4,  u4»». 

Ex. 

11. 

it 

1  0  t.nns_  AnR 

ka.  12 

00 

4 

xn 

6 

.    >,        ww--^, 

( 

i  firkms,  A7if 

El.  13. 


GREATEST    COMMON   DIVISOR 
^  Ex.  14.  3  U 

t  


21 


20  suits,  Ans. 


3 


115 

00  2 


230 


76|days,-4w«. 


GREATEST  COMMON  DIYISOR. 


(98,  page  74.) 


Ex.  1,     3  X4=:12,  ^?25. 

Ex.  3.    4x6  =  24,  Ans, 
Ex.  5.   2x7=1 14,  Ans. 
Ex.  7.   7x10  =  70,  ^?Z5. 
Ex.9.   2x7x9  =  126,^715. 
Ex.  12.  5x5  =  25,  Ans, 


Ex.2. 
Ex.  4. 
Ex.  6. 
Ex.8. 


2x3  =  6,  Am. 

3  X  6  =  18,  Ans. 

4  X  4  =  16,  Am. 

3  x5  x5  =  75.  Am, 


Ex.  10.4x8  =  32,  ^/i5. 


(OOj  page  78.) 

Ex.  8.  To  arrange  a  nnmber  of  things  in  equal  parcels,  the 
parcel  must  be  a  divisor  of  the  number ;  and  to  arrange  two 
numbers  in  equal  parcels,  the  parcel  must  be  a  common  divisor 
of  the  two  numbers.  Ans.  5  in  a  parcel. 

Ex.  9.  The  lots,  to  be  equal,  must  be  a  common  divisor  of 
the  three  fields,  and  to  be  the  greatest  possible,  must  be  the 
greatest  common  divisor.  Ans.  2  acres. 

Ex.  10.  To  avoid  mixing,  the  capacity  of  a  bin  must  be  a 
common  divisor  of  the  two  numbers  of  bushels ;  and  to  have 
the  least  number  of  bins,  will  require  the  yreatest  common  di- 
visor of  the  two  numbers  of  bushels.  Ans.  21  bushels. 

Ex.  11.  Tlie  greatest  common  divisor  of  124,  116,  and  92 
rods,  the  three  fronts.  Ans.  4  rods. 


22  PROPERTIES   OF   NUMBERS. 

Ex  12.  The  greatest  common  divisor  of  the  three  lengths, 
3013,  2231,  and  2047  feet.  Ans.  23  feet. 

Ex.  13.  The  greatest  common  divisor  of  the  three  num- 
bers of  bushels  is  2,  which  must  be  the  capacity  of  the  bag. 
Now  there  are  to  be  forwarded  2722-1-1822  +  1226  =  5770 
bushels;  and  5770-f-2zii:2885.  Ans,  2885  bags. 

Ex.  14.  The  greatest  common  divisor  of  $120,  $240,  and 
$384,  is  $24,  the  price  of  the  cows;  and  $120-t-$24=:5,  A's 
number;  $240-^$24  =  10  B's  number;  and  $384-r-$24  =  16, 
C's  number. 


MULTIPLES. 

(104,  page  81.) 

Ex.  2.  2  X  5  X  7  X  7=490,  Ans. 

Ex.  3.  2x2x2x3x7x  17  =  2856,  Ans. 

Ex.  4.  2x2x2x3  x  3  =  72,  Ans. 

Ex.  5.  2x2x2x3x5x7x11  =  9240,  Am. 

Ex.6.  2x3x3x5X5  =  450,  ^7Z5. 

Ex.  7.  2x2x3x3x5x  7  =  1260,  Ans. 


Ex.  4.  2,  3 

2,5,7 


(10^5  page  83.) 
42  ..  60 


7  ..  10 


2x2x3x5x  7=420,  Ans, 


Ex.  5.     3,  7 
2,5 


21  ..  35  ..  42 


5  ., 


8x7x2  X  5  =  210,  Ans, 


Ex.  6. 


Ex.  7. 


Ex.  S. 


Ex.  9 


Ex.  )'>. 


LEAST    COMMON   MULTIPLE. 


28 


2,5 
3,  2,  5,  5 


60  ..  100  ..  125 


6  ..  10 


25 


2x5x3x2x5x  5==1500,  Ans. 


2,2,5 

40  . 

.  96  ..  105 

2,  2,  2,  3 

2  . 

.24..  21 

1 

1 

2x2x5x2x2x2x3x  '7  =  3360,  An^ 


2,2,3 

48  . 

.60. 

.12 

2,  2,  3,  5 

4  : 

.  5  . 

.  6 

2x2x3x2x2x3x5  =  720,  Ans. 


2,2,3 

84  . 

.  224  . 

.  300 

7,5 

7  . 

.  56  . 

.  25 

2,  2,  2,  5 

8  . 

.   5 

2x2x3x7x5x2x2x2x  5  =  16800,  Ans. 


3,3 

270  ..  189 

.  297  . 

.  243 

3,  3,7 

30..  21. 

.  33  . 

.  27 

2,5,11,3 

10  .. 

11  . 

.   3 

3x3x3x3x7x2x5x11  x  3  =  187110,  Ans. 


Ex.  11. 


2,3,5 

5  . 

.6  . 

.7  . 

.  8  . 

.9 

2,  2,  1,  3 

, 

.  7. 

.4  . 

.3 

2x3x5x2x2x7x  3  =  2520,  Ans. 

Ex.  12.  To  purchase  books  at  5  dollars,  or  3  dollars,  or  4 
dollars,  or  6  dollars,  the  sum  of  money  must  be  some  common 
mvliiple  of  5,  3,  4  and  6  ;  and  the  least  sum  will  be  the  teasi 
iommor,  multiple^  which  is  60  dollars,  Ans, 

E.c.  13.  The  least  common  multiple  of  12,  15,  and  ly 
barrels,  whicb  is  180  barrels,  Ans. 

Ex.  14.  The  least  common  multiple  of  the  prices,  $30, 
$o5,  and  $105,  which  is  $2310,  Ans. 


24  FRACTIONS. 

Ex.  15.     The    least  common  multiple  of  41,  63,  and  64 

sheep,  which  is  15498  sheep,  Ans, 

Ex.  16.  He  must  spend  in  the  purchase  of  each  kind  of 
fowls  a  sum  equal  to  the  least  common  multiple  of  the  prices 
paid.  Suppose  he  takes  the  cheaper  turkeys  ;  the  least  com- 
mon multiple  of  12,  30,  and  75  is  300;  and  300^12=25, 
number  of  chickens;  300-^301=10,  number  of  ducks;  300 
-i-75  =  4,  number  of  turkeys;  and  25 -flO +  4  =  39,  the  whole 
number  of  fowls  purchased.  Next  suppose  he  takes  the  tur- 
keys at  the  higher  price;  the  least  common  multiple  of 
12,  30,  and  90  is  180;  and  1 80-^-12  ==15,  number  of  chickens  ; 
180-f-30  =  6,  number  of  ducks;  180-^90  =  2,  number  of  tur- 
keys; and  154-6  +  2  =  23,  whole  number  of  fowls.  But 
39  —  23  =  16,  number  of  fowls  purchased  more  than  was  n^ 
cessary;  and  16x5  =  80  cents,  Ans. 


FRACTIONS. 

(ISO,  page  88.) 

Ex.    2.  Ans,  /j, 

Ex.    4.  Ans,  ij. 

Ex.    6.  Ans,  JJ. 

Ex.    8.  Ans.  p^fy. 

Ey    10  An<f   — OJL^ 

Ex.11.  Nine  tenths;  seven  twelfths;  five  twentieths, 
jwelve  twenty-eights;  fifteen  seventy-fifths ;  nine  one  hun 
tf.red  twelfths;  forty-five  two  hundred  twentieths;  one  hup- 
dred  twenty -five /oz^r  hundred  twenty-eighths. 

Ex.  12.  Ninety  one  hundredths  ;  three  hundred  twenty- 
five  one  thousandths ;  four  hundred  fifty  one   thousand  tufo 


Ex. 

1. 

Ans. 

h 

Ex. 
Ex. 

3. 
5. 

Ans, 

A71S, 

Ex. 
Ex. 

7. 
9. 

Ans, 
Ans, 

Ts  6  0  0 

REDUCTION.  26 

hundred  fortieths  •  twenty-five  one  thousand  Jiie  hundredths  ; 
twelve  two  thousandths ;  seven  hundred  twenty-six  three 
thousand  four  hundred  seventy-fifths. 

Ex.  13.  Seventeen  one  hundred  fourths  ;  one  ten  thousand 
one  hundred  tenths ;  nine  hundred  fifteen  eighty-four  thousand 
six  hundred  twenty-firsts;  thirty-eight  thousand  sixty-five 
four  million  five  hundred  thirty-one  thousand  four  hundred 
twenty -ninths. 


REDUCTION. 

(126,  page  90.) 

Ex.    5.  Ifl^i,  Ans,  Ex.    6.    ^VtVt^/f^  ^ris. 

Ex.    Y.  tVYt^^  ^^^-  Ex.    8.   tV¥8=/3,  ^^*- 

Ex.  12.  fM=e.  -^^^-  Ex.  13.   i||=Ai  ^^^* 

Ex.  14.  \m=h  ^^^^ 

(127,  page  91.) 

Ex.  4.   A  jx:^  1532^  ^^5.  Ex.6,    ^p=5^i,  Ans. 

Ex.1.   ^8^=41,  ^715.  Ex.9.    3^2L— 430i|,  :247i«, 

(128,  page  92.) 

Ex.4.     140=:2.2.AA     ^^5.  Ex.  6.     94=8.fi     ^715. 

Ex.  7.    180:=-i^^|^  Ans,      Ex.  9.    247—^^31,  ^w^. 
(129.) 

Ex.  3.     7lf rrri-99,    ^715.  Ex.      5.  l^j^^^h^^    ■^^^' 

Ex.9.    96yVo==HM%  ^^«-     Ex.  11.  400||=-L4_8pL,^««. 


^,^: 


FRACTIONS. 


(130,  page  93.) 


Ex.2. 

15^5 

=  3; 

Ex.  3. 

l=A»  ^^^. 

Ex.4. 

51~1Y=3; 
if =jf,  Ans. 

Ex.  5. 

Ex.  6. 

3488—436=8; 

Ex.1. 

(ISSj 

page  95.) 

Ex.  2. 

5,5 
2,3 

50  . .  75 

2  ..    3 

35-4-7  =  5; 
f =11,  Ans. 

150-v-30=5; 

3  0 ISO*    -"'*'•»• 

1000—125=8; 
T2s~ToVir«  Ans, 


Ex.3. 


Ex  5. 
Ex.  7. 
Ex.  8. 
Ex.  9. 
Ex.  10. 
Ex.  11. 
Ex.  12. 


5  X  5  X  2  X  3  =  150,  least  common  denominator. 

^_     Ji       47        4    —    12  45        141      _1_        A^i* 


2,2,2 
2,3,7 


16  ..  21 


21 


2x2x2x2x3x  7=336,  least  com.  denom. 
±  4  _3_  JL— 15.1  1.3.3.  _6_3_  _3_2_     Ans 

2»T)    16)    21  —  336)    336»    336)    336>     -^'*^« 


Ex.  4.     3,  3,  2 


7,2 


9  ..21  ..4 


3x3x2x7x  2  =  252,  least  com.  denom. 

a.     _1_      3     3.  —  _5JB_     Jl_2_     J.19.     _1_5  1.5.       J770 
¥)    21)    4)    1  — 252)    252)    252)      252   »    -^'*''»» 

?JL  SL  JJ.— AA  j_a.  j_i.    Ans 

T)4)      8    —    8)      8)      8)    ■^'t'^' 

3  1     2     i_7     _5_ — 2.3. A    _aJ_    JL8_    ±l.ft    JLO        yl-w* 

4)  8')  T)     6   )    14  —  T6  8)    16  8)    16  T)    16  8)    168)    ^^^» 

4  7        115.     1 — 3i.     31      1_6_5      4JL5      35.       y|^o 

5)  TJ)      3)    1)    9—45)    4  5)      45   )     45    )    iJ)     ■^'*^« 

2  1    J  3.   1    11    6.  —  4.2.    i_o_4    45.    4 A    Vie     Ans 

2"  8")     T)    8)    14)1  56)      56)56»56)      56)     -^'*'^« 

2  1     31    A    5.    11    5  32.1    111    A 8.0     2Cr       4  4_       7J        ^yij 

10)40)   1)3)30)8  120)   120)   120)1^^*120)    I  t^l   ^"'' 


A     2       1      _7_- 
9)    3)   ¥)    12- 


7      5       1_1      1- 
fi)    7)      4)2" 


-1^     2  A     _ft_ 

'  36)    36)    36)36 

"56)    56)     16    )    5 


,  Ans. 


ADDITION.  27 

iliX.    Id.     ^f,    fo,    3,   -'j— fQ,   l^o,    3^,   f  0",     ^7i6. 

Fv     Ifi         7       2  5      _$        13      3 7       J2_5.18      140l2iI0       AnS^ 

UiX.    ID.     2  0>      4   >    10 »    1>    S>    2  — 2  0>      20    »    2  0")  "2  0-?    2  0»    2  0»    ^'^* 


ADDITION. 

(133,  page  96.) 

7  +  3+1+54-9     25     ^5      ^1      ^ 
Ex.2.     --±-^ =-=2-=2-,^.. 

4  +  5  +  7  +  1  +  3  +  1131        7 
^''•^'  12  -12^^2'  "^'''• 

^     ,      7  +  9  +  2  +  13  +  16  +  21     68     ^18     ^ 
Ex.4.     ^^ =.-=2-,Ans. 

41+63  +  71+89  +  109     373        13      , 

Ex.  5. = =:3 ,  Ans, 

120         120   120' 

^  ^   13  +  76  +  140  +  181+223  683   61   ^ 

Ex.6.  ■ ■ ■ ■ = — =2 — ,  Ans. 

225  225   75' 


(134,  page  97.) 

„  „   3  2   27  +  8  35  . 

Ex.  2.  T  +  -  = ■ — = — »  -4rwr. 

4  9    36    36' 

^  „   4  11  56+55  111    41   . 


^,3125 
^^•'^   4  +  8  +  7  +  12= 


126+214-48+70_265_,  97 

168      -168-^168'  "***'* 


18  FRACTIONS. 


14  .    9        2 


1274  +  2835  +  390  4499    404   , 

= ~1- ,  Ans, 

4095       4095   4095* 


42   9   Y   1 
^'-     li0+70  +  28+n= 


42  +  18  +  35  +  10  105  3  . 

--,  Ans 


140       140  4' 


^     ^      bl      131   24  1  2 
^^•''  Y5  +  160 +2^  +  2  +  3  = 

102  +  131  + 144  + 75 +  100_  17 
150         ""  25' 

„«      3124.56      78      9 
'^•«-     4  +  2  +  3  +  5  +  6  +  7  +  8  +  9  +  10  = 

1^90  +  1260  +  1680  +  2016  +  2100  +  2160  +  2205  +  2240  +  2268 

2520 

17819       ^179       , 


2520  2520' 


T.     ,^      4      9      2     19        ^19 
^^•^^-     5-^10  +  3  +  20=^60 
14  +  3+    1  =  18 


21U,  Ans. 


Ex.11.     f  +  T\  +  f=  2/j 
1  +  10  +  5  =  16 

18^,  -4/i«. 

Ex.  12.     f +  A  +  ,-V   =   lA 

17  +  18  +  26  =  61 

62/^,  Ans. 

Ex.  13.     /8+H  +  i  +  H  =  HJ 
1  +  3  =  4 

5^J,  -4««. 


SUBTRACTION.  2^ 

Ex.  14.    ;}+  t\+  1    =    l^T 

125  +  327  +  25— 477 

4782V,  -^^*- 
Ex.15.     in  +  U+H  +  ii-^\n=^Hih^'^' 

Ex.    16.      /o+H+T  +  To=      4 
3  +  2  +  40  +  10^:55 

55f ,  ^ws. 

Ex.  17.     f  +  f +  1         =     2/:, 
125  +  96+48=:269_ 

27I2V  yards,  Arts. 

Ex.  18.     |  +  ^  +  J=zlJ^ 

5  +  3  —  8 

9jV  yards,  Ans, 
Ex.  19.     ^V  +  i^  +  1  +  3  +  2        ^     311 
26+40  +  51+59  +  62  =  238 


241^f  acres,  Ans, 


Ex.  20.     |  +  4+±i  +  _7_  :=       23«, 

175  +  325  +  270  +  437  =  1207 


35 


1209/5  bushels. 

205  +  296  +  200  + 156  =   857 

$859||,  dollare. 


SUBTRACTION. 
(135,  page  99.) 


Ex.  2.     —-—=-=-,  Ans, 
9        93' 

14-11      3      1      . 
Ex.3.     -^-=^^=-,Ans. 


80  FRACTIONS. 


20  —  6      14     , 

Ex.5.     ^±Z?lJ^^,Ans. 

^     ^      YS-ll      64      1      . 
Ex.  6.  -— ^^~     ^7i^. 

128         128      2' 

» 

^     ^      182-110      72        6      ^ 

Ex.  7. =  —  =r — ,  Ans. 

348  348     29' 

(I365  page    99.) 

Ex.  2.     -— -=-^^= — ,  Ans, 
2     9       18        18' 

15     2     75-48      27       9 
^'''^'     24-5=~12Cr=T20  =  40'"^'**- 

^,34      51  —  32       19       . 

Ex.  4. -:=z =  — ,  Ans, 

8     17        136        136' 

^     ^        84       4      49-8     41      . 

Ex.  5.     ■ =  -i: — =ir-,  Ans, 

120     35        70        70' 

1500     50     125  —  100      25       . 
^^•'-     1728-7^==— 4-4-=U4'^"^- 

^  ^   60   332   720  —  83   637  , 

Ex.  7. — -= = ,  Ans, 

89  4272    1068    1068' 

Ex.  9.  8i  =  8y\        Ex.  10.  25|  =  25^^ 

4|f,  ^?Z5.  163^  =  16^3,^/1*. 

Ex.  11.  4f =4||        Ex.  12.  6 

3  jj,  ^725.  4-f,  ^715. 

Ex.  13.  450J  =450^1    Ex.  14.  ^^J  =  Z^^ 

i20Jv=i2oii  tVs=  m 

330jf,  Ans,  33Y5,  ^n». 


MULTIPLICATION.  81 

Ex.  15.  751                            Ex.  16.  227f 

49  1961=1961 

26i;  Ans.  30f ,  Ans. 

Ex.  19.  $Y|-~$6i=$lf^2,  Arts. 

Ex.  20.  4  +  31     =4V\  Ex.  21.  6i  +  2i+f=     9^^ 

5i-4i    =-  H  $255-$92«o  =$16Ht 

4_3_ - |i  =  3|f ,  ^n^.  -^^^*" 

Ex.  22.  n-2f =4|f,  ^7^5.  Ex.  23.  ^f-H=^»  ^^^«- 

Ex.  24.  9121  +  5451  =145^^ 

|2000-$145YyV=$542^^,  Ans. 

Ex.  25.  $136yV  +  $350|=$487|i  cost. 

$184i  +  $416J   =  $6011  receipts. 
$601i-$487ii=$llV8»  ^^^• 


MULTIPLICATION. 
(I375  page  101.) 


Ex.  4.  tV  X  '^=iT=lTf»  ^^^• 

Ex.  5.  1^4  X  12  =  VV='7f»  ^^^• 

Ex.  6.  /t  X  63=5  X  3  =  15,  ^W5. 

Ex.8.  '7|xl2  =  *|«=91i,  ^^5. 

Ex.  9.     tVt  X  8=f  It^SyVr,  ^^^• 
Ex.10,  ^l^x51=f  =  2,^r^5. 
Ex.11.  15fx  16  =  125  X  2=250,  ^w«. 
Ex.12.  mx22  =  'i'=lH,Ans. 
Ex.13.  $8Axl2=$Hf'=^106i,  ^n#. 

Ex.  14.  |iix9  =  $H=^S-I'  ^''^' 
Ex.  15.  $J  X  2'7=$H^=$23f,  ^w«. 


32 


Ex.    2. 


FRACTIONS. 

(138,  page  103.) 
Ex.    3. 


14 


100 
9 


450 


Ex.    4. 


21 


li,  Ans. 

105 
17 


Ex.    5. 


47 


85,  Ans, 


47 


64f,  Ans, 

19 
13 


247 


Ex.    7. 


42 
39 


Ex.    8. 


16 


5if,  Ans. 

80 
233 


819 


'1165,  Ans. 


409^,  Ans. 


Ex.    9. 


39 


156 

27 


Ex.  10.  $8xf  =  6|  dollars,  ^/MT. 
Ex.  11,  36  X  10|=384  miles.  Ana. 


108,  Ans. 
Ex.  12.     $450  X  ^2  =$262|,  Ans. 
Ex.  13.     $16  X  2J=$44^,  Ans. 

(139,  page  104.) 
I  Ex.    3.     8 


Ex.    2. 


10 


Ex.    4. 


24 
65 

10 


^,  Ans. 

11 
36 


Ex. 


5.     6 

7 


y\,  Ans. 


21 
6 

18 

3|,  Ans. 


MULTIPLICATION. 


33 


Ex. 

6. 

10 

9 

1 

2 

9 

5 

4 

1 

28 

1 

aV,  ^ns. 

E^.    9 


Ex.  11. 


Ex.  13. 


15 

8 

4 

9 

5 

1 

3 

22 

25 

44 

lif,  Arts. 


3 

2 

1 

n 

4 

5 

4 

4 

13 

35 

78 

2A,  -4^5. 


8 

7 

2 

1 

9 

11 

2 

3 

1 

8 

12 

77 

6y5_   Arts. 


Ex.  15      $ixf=$i  u4w5. 
Ex.17.     ix^=^J  Ans, 


:.  7.  6 

11 

6 

3 

2 

3 

16 

15 

176 

1111,  Ans. 


Ex.  10. 


7 

2 

1 

16 

10 

7 

3 

80 

3 

256 

851,  Am 


Ex.  12. 


2 

6 

4 

3 

5 

4 

3 

4 

2,  Ans. 


2 

25 

2 

11 

4 

27 

16 

7425 

Ex.  14. 


464i-V»    -^^** 

Ex.  16.     4x|=:J^,  ^W5. 
Ex.18.     V  x$|=$ff,  ^'/wj 


84 


Ex.20. 


4 
2 

51 
17 

8 

867 

FRACTIONS. 

Ex.  21. 


Ans.,  108f. 


8 
6 

51 

14 

20 

357 

Ans,^  11^1    dollai*. 


Ex.  22. 
Ex.  23. 
Ex.  24. 
Ex.  25. 


Ex. 
Ex. 
Ex. 


26. 

27. 
28. 


f  xf  x$VV=$21y\-,  Ans. 

V  x$V-=^22iJ,   Ans. 

f  X  V-  X I X  P^^-=$25^j,  Ans. 

Xx|=i4,  Ans. 

$A.^xi=$25-^\,Ans. 

■^^  acres  x  |  x  1=49-,-^  acres,  Ans^ 

f-f  X  f  barrels  =6|  barrels,  An>s. 


DIVISION. 
(I4O5  page  107.). 

Ex.    6.  j\^j---25z=^.l-^,Ans. 

Ex.    9.  $|-r-6=$i,  Ans.        Ex.  10.     J-v-7=|-,  ^7i5. 

Ex.  11.  -f  -v-5  =  3«j,  -47^5.  Ex.  12.     $^-^-9=$^,  Ans. 

Ex.  14.  V-v-3  =  5yV  ^^5-      Ex.  15.     ;2  x  Y-^^ -|,^w*, 

Ex.  16.  $2.±|iL_^4  =  $\2_4._$24||,  ^7^s. 

(141 5  page  109.) 

Ex,    7.  I X  9-^1=15,  ^715. 

Ex.    8.  121-x-^— $49,  u4w5. 

Ex.    9.  16xf  =  10;  10-^|-=:22i,  .^WA. 

Ex.  11.  75-i-Y-=^H»  •^^^• 

Ex.  12.  149-^J4i=6yVj,  ^w«- 

Ex.  18.  15^-f =9,  Ans. 


Ill  VISION. 


B6 


Ev  14.     f  X  320=200 ;  f  x  Y=¥ ; 

200— V =254,  Ans. 
Ex.15.     $32x1=8;  Yxi=f; 

Ex.  16.     183~J^i=4,  Ans. 


Ex.  2. 


Ex.  4. 


Ex.  6. 


Ex.  8. 


Ex.  10. 


(143, 

page  110.] 

) 

8 

7 

Ex.3. 

9 

6 

3 

4 

1 

6 

6 

r" 

3 

10 

1 1^,  Ans. 

31,  Ans. 

1 

4 

Ex.  5. 

2 

1 

9 

10 

7 
14 

13 

63 

40 

13 

ih  ^^«- 

\h  -4^*' 

3 

2 

Ex.7. 

6 

5 

27 

28 

4 

5 

81 

56 

• 

24 

25 

ih  ^ns. 

l^V,  ^^*' 

3 

5 

Ex.  9. 

19 

17 

8 

7 

7 

19 

9 

35 

7 

17 

3f ,  Ans. 

2f ,  ^»«. 

20 

13 

Ex.  11. 

7 

2 

5 

16 

2 

6 

25 

52 

1 

3 

2 
4 

2^,  ^»5. 

21 

40 

l^f,  Ans. 


FRACTIOOTL 


Ex.  12. 


10 

9 

4 

5 

6 

13 

4 

325 

432 

6f     56      3      28     . 
Ex.  15.     -^=— -  X  TTTT^rrr,  Arts. 
81       9      26     39' 


Ex.  16. 
Ex.  17. 
Ex.  18. 
Ex.  19. 


"'a 


80     1 


— ^=— -x-=20,  Ans. 
4        Y      4         ' 

^5        5       5       25       . 
ii= — X — = — ,  Ans* 
4J     11     22     242' 

_-x-x--l,^n., 


2     5     9     2     1. 
— -  X  -  X  -  X  -=-,  Ans. 
|xf5629     3' 


2   X  A 


Ex.  20.     V  x|=14,  Ans. 

Ex.  21.     3_3  X  f = Y=6|,  ^M5. 

Ex.  22.      8     35  •     Ex.  23. 


8 

35 

1 

2 

6 

6 

2 

21 

3 
14 


98 
6 

35 
'uT^Ans. 


Ex.24.     $J43.xixf=$14f ; 

$17— $141=$  2^,  Ans. 

Ex.  25 


10 
2 

20 


$/^,  Ans. 


Ex.  26. 


3 

10 


PROMISCUOUS   EXAMPLES. 
Ex.  27.     1 


87 


10 

2 

3 


2  bu.,  Arts. 


10 


1905 
8 


127 


12j-V,  Ans 


PROMISCUOUS   EXAMPLES, 

(Page  112.) 

Ex.2.     91^7  =  13;  4^;  1 3=f^,  ^w*. 
Ex.  3.     3,  40  I  3  ::  40 

3x40  =  120,  Ans, 

Ex.  4.     4  +  3  =7 

i  +  J  +  f  of  f  =  2|f|_ 


Ex.6.     |xn=fi=4J, 

li|},  Ans. 
Ex.  6.     47561- +  1281=4885 3V,  ^w«. 

Ex.7,     f  xix|xY-=H=VTr 

Yxfxfx  |=H=3li_ 

31J,  ^ris. 

Ex.  8.     f  X  f =20,  ^W5. ;  f  x  f = V  =1t»  ^^*- 

Ex.  9.     18251=1^1-^;  iAjiLixf=^4|i^=3043i,  .4rw. 

Ex.  10.  i+i=/o  ;  1-2V=H;  ^V-i^  =  140,  ^715. 

Ex.  11.  ^  X  $  V  X  V  =^24^1,  ^ws. 

Ex.  12.  $V  X  |-=$23i,  ^715. 


88 


Ex   13. 


Ex.  15. 
Ex.  16. 

Ex.  17. 

Ex.  18. 
Ex.  19. 
Ex.  20. 
Ex.  21. 
Ex.  22. 
Ex.  23. 
Ex.  24. 

Ex.  25. 
Ex.  26. 

Ex.  27. 


FRACTIONS. 

8 
14701 

14701               Ex.  14.        8 
2                                  471 

37803 
2 

4 

1 

628 

$12601,  Ans, 

$1,  Ans. 

P-Y^  X 

^  Xf =$40551,  ^ws. 

1     42      10      2,22,, 


^*1  ( 
y  x|xf=27,  Ans. 

V  X  V  X  l=¥/=24-i-V,  ^ns. 

V  xf  x|=34i,  ^715. 
lx\^=^=1j\,Ans. 

3_3  X  _3_  xifi=iy-a=589|,  ^W5, 
H^  X  yaVo  X  ^V^=2500,  Ans. 

$204- f  =$50,  ^715. 
4         5  —  20J-'-         20  —  20?     ^^'20  — ^^)  -n/«^, 

$i7^j.Xf=$4608,  Ans. 

320  X $21=$  720  755x|X$lf=   528^ 


435x$lJ=:$  815f 


755x|x$2i-: 


755      $1535| 
$15351— $1491^ 


962f 
$14911 


l:^^,  Ans. 


^     ^^    14-5     12  12  7   5   , 

£x.  28.  -  -= — : = — ,  Ans. 

8  +  5  13'  13  8  104' 

_  ^^  8  +  5  13  8  13   6   ^ 
Ex.  29.  J-     ,=— -;  71— —  =ri,  Ans. 
7  +  5  12'  7  12  84' 

Ex.  30.  f  X  f  X  f  X  |=7-J,  Ans. 

Ex.  31.  J  X  $V  X  f  X  ^=  V  =$3J,  Ans. 


DIVISION.  39 

Ex    32.  16f-3J  =  12l=i?/ 

H^x'V  X  2%=^  W^  =95311,  Ans. 
Ex.33.  -Vxy  =  'LV  =  12ff, -4w5. 
Ex.  34.  6-^-1846=^1^,  Ans. 
Ex  35.  $V  X  f  X  3  J =$3,  ^W5. 
Ex  36.  $1-6  X  f  X  i  X  J  X  f-=$5,  ^W5. 

Ex     SY      5  4-J'-  — 17.     1  11  — _3_-    _3 1—1 

20-^^V  =  800,  ^715. 
Ex.  38.  Jf  cents  xy  xf  =  100  cents,  Ans. 
Ex.  39.  i+f =H  ;  1-  H  =  2T,  remainder. 

^f i  X  2^  X  $2.1^=^1  i-^=$45YYW,  ^^^• 

Ex.  40.     If  the  horse  cost  1\  times  as  much  as  the  wagon, 
ihe  horse  and  wagon  must  cost  2  J  times  the  wagon.      Hence, 
$270-T-2i=$120,  Ans. 

Ex.41.     Y^xf=32;  32-20f  =  lli,  ^W5. 

Ex.  42.     P^-i^  X  aV  X  $1=126,  Ans, 

Ex.  43.  If  A  can  do  f  as  much  as  B,  he  can  do  the  work 
in  4.  of  the  time  that  B  will  require,  and  in  1+|=|-  of  the 
time  be  Ah  will  require.     Hence 

14    days  x  |=32|  days,  A's  time ;  ) 
32|  days  x  J=24i  days,  B's  time ;  )      ^* 

Ex.  >  4.     -V^  X  }  X  f =11  J,  Ans. 

Ex.  45.     A,  B,  and  C  can  do  J  of  the  work  in  a  day ; 

B  and  C  can  do  \  of  the  work  in  a  day ;  hence 
A  alone  can  do  \--\z=:^^  of  it  in  a  day ;  and 
be  wiP  therefore  require  ^^=:1Z\  days,  Ans. 

fix.  46.     1  +  1+1=/^;     1-^^=J^,  remainder; 
tV-tV=3V;  $24~3V=$720,  Ans. 

Ex  11     -V^  X  aV  X  V  =Hh  ^^5- 


10 


DECIMALS. 


Ex.  48. 
Ex.  49. 
Ex.  50. 
Ex.  51. 


Ex.52. 


i-i=j\  ;  30  feet-^y3__ioo  feet,  Ans, 

i+  i  =T2>  fraction  of  the  post  below  water, 
1-t'2=tV         "  "  "     (^^ove     " 

21-h/^=36  feet,  Ans. 

^=  eldest  son's  fraction; 
^x^=l^=  youngest  son's  fraction; 
1  —  (-?  +  If)  ==  Jf  =  daughter's  fraction ; 
n-if  =  4V;  ll'723f--/^=$21}U;f  ^Twr. 


DECIMAL  FRACTIONS. 


(145,  page  118.) 


Ex.  1. 

Ans,  .38. 

Ex.  2.   Ans     (. 

Ex.  3. 

Ans.  .325. 

Ex.  4.   Ans     04. 

Ex.  5. 

Ans.  .016. 

Ex.  6.   Ans.  .00074. 

Ex.  7. 

Ans.  .000745. 

Ex.  8.    Ans.  .4232. 

Ex.9. 

Ans.  .500000. 

Ex.  10 

.      Five  hundredths; 

twenty-four  hundredths;    six 

hundred  seventy-two  thousandths;  six  hundred  eighty-one 
thousandths ;  twenty-four  thousandths;  eight  thousand  four  and 
seventy-one  ten-thousandths;  nine  thousand  thirty-four  ten- 
thousandths;  five  ten-thousands  ;  one  hundred  thousand  two 
hundred  forty-eight  millionths ;  nineteen  thousand  two  hun- 
dred forty-eight  hundred- thousandths ;  opc  thou&dnd  three 
hundred  eighty-five  millionths  ;  one  million  eighty-pere^  t^D- 
millionths. 


NOTATION   AND   NUMERATION.  41 

(I465  page  118.) 


Ex.  1.    Ans, 

18.027. 

Ex.  2.   Ans.  400.0000019. 

Ex.  3.   Ans. 

54.000054. 

Ex.  4.  Ans.  81.0001. 

Ex.  5.   Ans» 

100.0067. 

Ex.  6.  Eighteen,  and  twenty-seven  thousandths;  eighty* 
one,  and  one  ten-thousandth ;  seventy-five,  and  seventy-fiv<, 
thousandths ;  one  hundred,  and  sixty-seven  ten-thousandths ; 
fifty-four,  and  fifty-four  millionths ;  nine,  and  two  thousand 
eight  hundred  six  ten-thousandths ;  four  hundred,  and  nine- 
teen ten-millionths ;  three,  and  three  hundredths ;  forty,  and 
forty  thousand  four  hundred  four  hundred-thousandths. 

(148,  page  120.) 

Ex.    1.   Ans.  ,000^25,  Ex.    2.   Ans.  .6000. 

Ex.    3.   Ans.  .01859.  Ex.    4.    Ans.  .000260008. 

Ex.  5.  Six  thousand  three  hundred  twenty-one  ten  thous- 
anths ;  five  million  four  hundred  thousand  twenty-seven  ten- 
millionths  ;  seven  hundred  forty-eight  thousand  two  hundred 
forty-three  millionths ;  sixty  million  hundred-millionths  ;  two 
million  nine  hundred  sixty-two  thousand  nine  hundred  ninety- 
nine  ten-millionths ;  six  hundred-millionths. 

Ex.    6.   Ans.  502.001006.         Ex.    7.   Ans.  31.0000002 

Ex.    8.   Ans.  11000.00011. 

Ex.    9.     Ans.  9000000.000000009. 

Ex.  10.   Ans.  10.2.  Ex.  11.   Ans.  124.315. 

Ex.  12.  Ans.  .700.  Ex.  13.  Ans.  .00007. 

Ex.  14.  Twelve,  and  thirty-six  hundredths ;  one  Lundred 
forty-two,  and  eight  hundred  forty-seven  thousanths ;  one,  and 
two  hundredths ;  nine,  and  fifty-two  thousandths  ;  thirty-two, 
and  four  thousandths ;  four,  and  five  ten-thousandths ;  six- 
ty-two and  nine  thousand  nine  hundred  ninety-nine  ten- 
thousandths  ;  one  thousand  eight  hundred  fifty-eight,  and  four 


^2  DECIMALS. 

thousand  five  huadred  eighty-three  ten-thousandths ;   twenty 
seven,  and  forty-five  hundred-thousandths. 


REDUCTION. 

(149, 

page  121 

0 

Ex.  2.       .I'ZOOOOO 

Ex.3. 

.'TOOOOO 

24.6000000 

.024000 

.0003000 

.000187 

84.0000000 

.000500 

721.80002'71 

108.450000 

Ex.  4.     1000.001000 

841.'780000 

2.600400 

90.000009 

6000.000000 

(150,  page  122.) 
Ex.  2.   TV¥o=i»  ^^«-  Ex.  3.    tVo  =  2V»  ^^«- 

Ex.4.   jWo=m^^^^'  Ex.5.   y»JV¥o  =H» -4^- 

Ex.6.   j^i^^=^^\^^  Arts. 

(151,  page  123.) 
Ex.    4.   Arts.  .4.  Ex.    6.   Ans.  .875. 

Ex.    9.   Ans.  .375.  Ex.  10.   Ans.  .0375. 


ADDITION. 
(152,    page  124.) 
Ex,  6.  26.26  Ex.    7.     36.015 

.7  300.0605 

6.083  5.000003 

4.004  60.0000087 


87  047,  Ans.  401.0755117,  Ans, 


ADDITION. 


43 


Ex.    8. 


64.34 
1.0009 
3.000207 

.023 
8.9 
4.135 

71.399107;  Ans. 


Ex. 


9.  18.375 
41.625 
35.5 


95.500,  Ans. 


Ex.  10 


61.843 
143.75 
218.4375 

21.9 


Ex.  11. 


445.9305,  Ans. 


Ex.  12. 


^= 

2.5 

5f  = 

5.75 

H= 

3.625 

3.0642 

8.925 

23.8642  barrels,  Ans. 


12J   =12.75 

18f   =18.4 
9=9 

241   =24.125 
4|f=   4.8125 
8yV=   8.9 

15iJ=15.65 


93.6375,  Ans, 


Broadcloth. 
Ex.  13.     First    suit,      2.125 
Second "         2.25 
Third    "         5.0625 


Sums 
Total 


Cassimere. 

Satin. 

3.0625 

.875 

2.875 

1.000 

1.125 

9.4375  5.9375  3.000 

9.4375  +  5.9375+3  =  18.375,  Ans, 


44 


DECIMALS. 


SUBTRACTION. 


(I535  Pag^  126.) 


Ex. 

4. 
6. 

8. 

10. 
12. 

714.000 
.916 

Ex.  5. 
Ex.  7. 
Ex.  9. 

Ex.  11. 

2.000 
.298 

Ex. 

713.084,  Ans. 

21.004 
.75 

1.702,  Ans. 

10.0302 
.0002 

Ex. 

20.254,  Ans, 

900. 
.009 

10.03,  Ans 

2000. 

.002 

Ex. 

899.991,  Ans. 

1. 

.000001 

1999.998,  Ans. 

.427 
.000427 

Ex. 

.999999,  Ans. 

.34 
.034 

.426573,  Ans. 

.306,  Ans. 

MULTIPLICATION. 


\  (1*3^4:5  page  127.) 

Ex.  4.     274.855,  Ans.  Ex.  8.     243.6,  Ans. 

Ex.  12.  .000030624,  Ans. 


NOTATION   AND   NUMBKATION.  46 

\ 

DIVISION. 

(15>5,  page  129.) 

Ex.    5.  .111,  Ans.  Ex.    6.  11.1,  Ans. 

Ex.    8.  15,21 -\-,  Ans.  Ex.    9.  1;  10;  100;  1000,  ^ws, 

Ex.10.  5,68Ui-,Ans.  Ex.12.  3020,^^5. 

Ex.  17.  3.65,  Ans. 

PROMISCUOUS   EXAMPLES. 

(Page  130.) 

Ex.  2.     6188.311478,  Ans.      Ex.  3.    86.913,  Ans. 

Ex.  6.     .00012,  Ans.  Ex.  9.   4,  Ans. 

Ex.  11.  70.6755-T-6.35r^ll.l3,  Ans. 

Ex.  12.  tWo -f,  ^ns.  Ex.  13.    26yVVo=26i,  Am, 

^■'''^^=-''^^- 

^     ,^    .25x17.5      ^     . 

Ex.  16.  —-—=5,  Ans. 

.5x1.75 

Ex.  17.    3.625  X  36.75  x  $.85=$113.2359375,  Ans. 

Ex.18.   56.925-^4.6  =  12.375  =  12f,  ^n«. 


DECIMAL  CURRENCY. 

NOTATION  AND  NUMERATION. 

(I6O5  page  132.) 

Ex.  2.   Ans.  $2.09.  Ex.  3.   Ans.  $10.10. 

Ex.  6.   Ans.  $.032.  Ex.  7.   Ans.  $100,011 


46  DECIMAL   CURRENCY. 

Ex.  8.  Seven  dollars  ninety-three  cents/;  eight  dollars  two 
cents  ;  six  dollars  fifty-four  cents  two  mills. 

Ex.  9.  Five  dollars  twenty-seven  cents  two  mills ;  ono 
hundred  dollars  two  cents  five  mills;  seventeen  dollars  five 
mills. 

Ex.  10.  Sixteen  dollars  twenty  cents  five  mills ;  two  hun- 
dred  fifteen  dollars  eight  cents  one  mill ;  one  thousand  dol- 
hrs  one  cent  one  mill ;  four  dollars  two  mills. 


REDUCTION. 
(161,  page  133.) 

Ex.  2.  Ans.  3600  cents.  Ex.  3.  Ans.  524800  cents. 

Ex.  6.  Ans.  160  mills.  Ex.  1.  Ans.  3008  mills. 

Ex.  8.  Ans.  890  mills. 

(162,  page  134.) 

Ex.  2.  Ans.  $15.04.  Ex.  3.  Ans.  $138.75. 

Ex.  4.  Ans.  $16,525.  Ex.  5.  Ans.  52.4  cents. 

Ex.  6.  Ans.  $6,524. 


ADDITION. 

(163, 

page  134 

.) 

Ex.  2.  $     50.07 

Ex. 

3. 

$  364.541 

1000.75 

486.06 

60.003 

93.009 

.184 

1742.80 

1.01 

Ans. 

3.276 

25.458 

$2689.686,  - 

$1137.475, 

ADDITION. 

Ex.  1  $  .92  Ex.  5.  $89.74 

.104  13.03 

.357  6.375 

.186  19.625 

.125 
.99 


$ai2e  Ans. 

Ex.  6.  $  9.17  Ex.  7.  $2175.75 

.875  240.375 

.0625  605.40 

.04  140.125 
.08 
.11 


$3161.65,  Ans. 


$10.3375,  Ans. 


Ex.  8.  $  6.08  Ex.  9.  $7425.50 

26.625  253.96 

16.000  170.09 

7.40 


156.105,  Ans. 


$7849.55,  Ans. 


Ex.  10.  $3,625 

1.75 

1.375 

.625 

.875 


$8.25,  Ans. 


4^  DECIMAL   CURRENCY. 


SUBTRACTION. 

(164 

5  page  136.) 

Ex.  2.  $365,005 

Ex.  3.  $50. 

267.018 

.60 

$97,987,  Ans. 

$49.50,  Ans. 

Ex.  4.  $100. 

Ex.  5.  $1000. 

.001 

.037 

$  99.999,  Ans. 

$  999.963,  Ans. 

Ex.  6.  $1834.16 

Ex.  7.  $145.27 

1575.24 

37.69 

$  258.92,  Ans. 

$107.58,  Ans. 

Ex.  8.  $6.84 

Ex.  9.  $14725 

5.625           $3560-|-$'3^015.875=:10575.876 

$1,215,  Ans. 

$4149.125,  Ans 

Ex.  10.                $13.75 

5.25 

1.375 

.875 

$25-$21.25=: 

$3.75,  Ans. 

Ex.  11.                $480 

80.50 

$560.50- 

-$200=$360.50,  Ans. 

MULTIPLICATIOK 

(16^5  page  137.) 

Ex.  2.     $4,275  X  300=:$1282.50,  Ans. 
Ex.  3.     $2.45  X  176=:$428.75,  Ans. 


DIVISION.  49 


Ex.  4.     11.28  X  800:=$1024,  Ans. 

Ex.5.  $.15  x3Y2=:$55.80 
.125x434=  54.25 
.33    X    16=     5.28 


$115.33,  Ans. 

Ex.  6.  $.56  X  3=:$1.68 
.07x15=  1.05 
.08x27=   2.16 


$5--$4.89  =  $.ll,  Ans. 

Ex.  7.     $.375  X  125  =$46,875 

.09    X    75  =  $6.75 

.60    X    12=   7.20  =   13.95 


$32,925,  Ans. 


Ex.8.     $32.50  X  80  r 

34.25x70=   2397.50 


$4997.50 
3975 

$1022.50,  Ans. 


DIVISION. 

(166,  page  138.) 

Ex.  2.  $41.25-^33=$1.25,  Ans. 

Ex.  3.  $94.50-^27=$3.50,  ^715. 

Ex.  4.  $136-h64  =  $2.125,  Ans. 

Ex.  5.  $1.32-^$.12  =  11,  Ans. 

Ex.  6.  $405-f-$.54  =  750,  Ans. 

Ex.  7.  $180-^12=$15,  Ans. 

Ex.  8.  $2847.50 -M00=$28.475, -4715. 

Ex  9.  $80.46-j-894=$.09.  Arts. 

Ex.  10.  $1,125  x  120=$135  ;  $135-4-27=$5,  Ans. 

K.P,  3 


so  DECIMAL   CUiaiENCY. 

Ex.  11.      $3.20  X    4=:$12.80 
.08x37=     2.96 


$15.76 
6.80 


$8.96-^$.16=56,  Ans. 

Ex.  12.     $4.50  4- $2.75 =$7.25  ; 

$166.75-^$7.25  =  23,  Ans, 
Ex.  13.     $18.48-M54=:$.12,  Ans. 
Ex.  14.     $560 

106.75 


$453.25-M4=$32.37i,  Ans. 


ADDITIONAL  APPLICATIONa 

(I685  page  139.) 

Ex.2.  693x$i=$321,  ^7i6\   Ex.3.   478x^=$2S9,  Ans. 

Ex.  4.  4266  X  $yV=$355.50,  Ans, 

Ex.  5.  1250  X  $i=$156.25,  Ans, 

Ex.  6.  3126  X  $yV=$195.375,  Ans. 

Ex.  7.  1935  X  $i=$322.50,  Ans. 

Ex.  8.  56480  x  $^=$7060,  Ans. 

Ex.  9.  1275  X  $i=$255,  Ans, 

(169,  page  140.) 

Ex.  2.  $.09  X  864=177.76,  Ans. 

Ex.  3.  $1.25  X  87=$108.75,  Ans. 

Ex.  4.  $1.45  X  400  =  1580,  Ans, 

Ex.  5.  $.44  X  52  X  16=$366.08,  Ans. 

(170,  page  141.) 

Ex.  2.     $l75-4-25=$7,  Ans, 
Ex.  3.     $200-^48=$4.16|,  ^/^^. 


ADDITIONAL    APPLICATIONS.  51 

Ex.  4.  $1200-f-96  =  $12.50,  Ans. 

Ex.  5.  $56.25-f-10  =  $5.62J,  Arts. 

Ex.  6.  m.^O-MS^r^-GS,  Ans. 

Ex.  V.  llO.O'Z-f-SS^S.lQ,  Ans. 

Ex.  8.  $1016-T-800  =  $1.2Y,  Ans. 

Ex.  9.  $8'74.65-^343=$2.55,  Ans. 

Ex.  10.  $684.3'75-^36o=:$1.8'75,  Ans. 

(in,  page  142.) 

Ex.  2.  $5.55-f-$.15=37,  ^lw5. 

Ex.  3.  $2 16  ^$12  — 18,  Ans. 

Ex.  4.  $21Y8.'75-^$1.25  =  1'743,  ^W5. 

Ex.  5.  $643.50^$19.5=ir33,  ^?^5. 

Ex.  6.  $52.65-f-$.45  =  ll'7,  ^?i5. 

Ex.  Y.  $6336-f-$132=:48,  ^n5.  ^ 

Ex.  8.  $117715-^165  =  1811,  ^w«. 

(ITS,  page  143.) 

Ex.  2.  $4.50  X  42.65  =  $191.925,  Ans. 

Ex.  3.  $.85  X  24.89  =  $21.156.+,  Ans. 

Ex.  4.  $17.25  X  7.842=$135.274+,  ^w«. 

Ex.  5.  $12.50  X  23.48 =$293.50,  Ans. 

Ex.  6.  $3  X  1.728=$5.184,  Aiis. 

Ex.  7.     $7      X    2.40  =$16.80 
6.40  X      .865=     4.671 
.80X12.56   =   10.048 

$31,519,  Alls. 
Ex.  8.     $4,375  X  14.76 =$64,575,  Ans. 

(173,  page  144.) 

Ex.2.     $7-4-2  =  $3.50; 

$3.50  X  1.495=$5.2826,  Ans. 


52  DECIMAL    CURRENCY. 

Ex.  3.     $S.75~-2=z$4.Sl5; 

$4,375  X  .325  =  $1.421  -f ,  Ans. 
Ex.  i.     $3.84-^2=$1.92; 

$1.92  X  3.142=$6.032  +,  An8. 
Ex.  5.     $5.60-^2=$2.80; 

$2.80  X  1.848  =  $5.1744,  Ans. 

Ex.  e     $18-^2=$9; 

$9  X  125  X  .148=r:$33.30,  Ans. 
Ex.  7      $3.054-2r:r$1.525; 

$1,525  X  31.640=$48.251,  Ana, 

(174.5  page  145.) 

Ex.  1.     $3.60    X    '7=r$25.20 

1.125  X    9=   10.125 

•.90    xl2=   10.80 

1.375x24=    33.00 

.65    x32=:   20.80 


Ex.  2. 


Ex.  3. 


$99,925, 

Ans. 

$3.75 

X    67  = 

:$251.25 

2.62 

xl08  = 

:   282.96 

1.12 

X    75  = 

:      84.00 

.86 

X    27  = 

:     23.22 

.70 

X    35  = 

:      24.50 

1.04 

X    50  = 

:      52.00 

$717.93, 

Am. 

$.07 

X325: 

=$22.75 

.0625x148: 

=     9.25 

.05 

X286: 

=   14.30 

.125 

X    95  = 

=   11.875 

2.75 

X     50: 

=  1-37.60 

3.625 

X     75: 

=  271.875 

2.85 

X     12r 

=   34.20 

$501.75,  Ans. 


PROMISCUOUS  EXxiMPLES.  ti 

Ex.  4.  $15    X  20  =8300. 

9.50  X  7.5         =   71.25 
C.25  X  10.75       =  C7.1875 
2.625  X  3.96        =   10.395 
3.00  X  5.287       =  15.801 


$464.6935,  Atu 


Ex.6.   $.11   X  25=12.75 
.625  X  5=   3.125 
.0625x26=:  1.625 


.42 

X    4=   1.68 

.09 

x46=   4.14 

.14 

x30=  4.20 

.04 

X    6=     .24 

.12 

X    4=     .48 

$18.24,  Ans. 

PEOMISCUOUS  EXAMPLES. 
(Page  146.) 

Ex.  1  $124.35  X  62.75=$7802.9625,  Am. 

Ex.  2.  $.17x15=82.55,  ^?i5. 

Ex.  3.  $1406.25-v-2250=$f,  Arts. 

Ex.  4.  $48.96-M2=$4.08,  Ans.         ^ 

Ex.  5.  325  miles  x  .45  =  146.25  miles,  Ans, 

Ex.  6.  657-^36.5  =  18,^725. 

Ex.  7  $105+$125  +  ($35x4)=$370 

$400  — $370     =$30,  Ans. 

Ex.  8.  $19— $15  =  $4;  $4x28=$112,  ^?i«. 

Ex.  9.  ■V-X2\  =  ¥-='^A  •^^^• 
Ex.  10.  $9-^$.3125  =  28.8,  Ans. 
Ex.  11.     $3.50  X  365=$1277.50 

$2000-1277.50  =  $722.50,  ^/w?. 


M  DECIMALS. 

Ex.  12.       $687.25 +  $943.64=:$1G30.89 

$1630.89  — $875.29  =  $755.60,  Ans. 
Ex.  13      $l728-7-2=:$864  1st  half  sold  for; 
144x8  =  1152  2d     "       "     " 


$2016,^725. 

Ex.  14.     $3.75  X  .875  =  13.281  +,  Ans. 

Ex.  15.     $65.42  — $46.56  =  118.86,  gain  per  head; 

$3526.82-^-$18.86  =  187,  Ans, 
Ex.  16.     $54.72-^36.48  =  $1.50  ; 

$1.50  X  14.25  =  821.375,  Ans. 
Ex.  17.     $3548-^4  =  $887,  Ans. 
Ex.18.     112.34-^$.82  =  137,  ^n5. 
Ex.  19.     $3461.50-^46  =  $75.25  ; 

$75.25  X  5  =  $386.25,  A718. 
Ex.  20.     $24000  X  |  x  j  =  $3200,  Ans. 
Ex.21.     $1.25x160=   $200 

$5       X    80=     400 


$600 
$2.50x240=     600 


Loss      000,  Ans, 

Ex.  22.     $1.70x48  =  881.60 
72.90 


$  8.70,  Ans. 

Ex.23.     122|4-75i  =  197|;  197f — 60  =  137f  ; 

$.9375  — $.8125  =  $.125,  loss  per  bushel; 
$.125  xl37f  =  $17.218 -f     loss; 
12.50  gain; 

$4,718  +  ,  loss,  Ans. 

Ex  24.      $1.40  X  e =$8.40  wages  ; 
$  .75x7=  5.25  expenses, 

$3.15  savings,  Ans. 


PROMISCUOUS   EXAMPLES.  55 


$.08  X  39 
Ex.25,     -jj^ =19^,  Ans. 

Ex.  26.     $4.50  X  23.487=:$105.6915,  Ans. 
Ex.  27      $1200-^365:=z$3.2874f,  Ans. 
Ek  28.     $.17  X  56  X  28  =  1266.56,  ^W5. 
Ex   29.     $.07  X  26  X  13  X  16=$378.56,  Ans, 
Ex.  30.     $4.75  X  4.868=$23.123,  Ans. 
Ex.  31.     $.33ix27  =  $9.00 

.25    x28=  7.00 

.50    xl9=  9.50 


$25.50,  Ans. 
Ex.32.     44—32  =  12; 

— - — -=21^  minutes,  Ans, 
12  ^  ' 

$32.3      4      15     ^ 
Ex.  33.     — --  X  —  X  —=$51,  Ans. 
J         It/      z 

Ex  34.     $5.d35-f-.875=$6.44;   $6.44  x  9^ =$59.57,  Ans. 

Ex.  35.  $5000 

$1200.25  x3  =  $3600.75 
1800.62x8=   5401.86 


$14002.61 
$950.87x2=     1901.74 


$12100.87,  Ans. 
Ex  36.     $4.50xl86.40=$838.80,  J7i5. 

Ex.  SS.     $96.40-v-2=$48.20  ; 

$48.20  X  1.375=$66.275,  Ans. 
Ex.39.     ,Vo¥t=JL,  ^W5 


5Q  COMPOUND    NUMBERS. 

Ex.40.     3\=.09375;  .62i=.625;  .37y^  ::^,370625  ; 

|=::.375  ; 

.09375  +  .625  +  .370625  +  .375  =  1.464375,  Ans. 

Dr.  Cr. 

Ex.  41.       $4,745  $2,765 

2.625  1.245 

1.27  .625 

.45  3.45 

5.285  1.875 


Ex.  42. 


$14,375         -           $9.9C 

$.125  X  120=115.00 
.625  X    18=   11.25 
.07    X    47=     3.29 
.18    X      6=     1.08 

J=$4.415,  Ans. 

$1.50 
1.27 
1.87 
2.30 

$30.62   - 

$6.94 =$23.68, 

•    REDUCTION. 

(183,  page  152.) 

Ex.  1.     14194  far.-^4  =  3548  d.  2  far. ;  3548  d.-^12 
=  295  s.  8  d. ;  295  s.-f-20=14  £  15  s. 
Ans,  14  £  15  s.  8  d.  2  far. 

Ex.  2.     14£x  20  +  15  s.=295s.;  295s.  xl2  +  8d.=3548d,: 
3548d.x4  +  2  far.  =  14194  far.,  Ans. 

Ex.3.     15  £x  20  +  19  s.  =  319  s.;  319  s.xl2  +  lld. 

=  3839  d. ;    3839  d.x  44-3  far.=  15359 'far.,  Ans 

Ex.  4      15395  far.-~4  =  3839  d.  3  far.;  3839  d.-7-12 
=  319  s.  11  d. ;  319  s.-v-20  =  15£  19  s. 
Ans.  15  £19  s.  11  d.  3  far. 

Ex.5.     46  sov.x  20  +  12  s.  =  932s.;  932s.xl2+2d. 
=  11186  d.,  Ans 


REDUCTION.  57 

Ex.  6.  11186  d.-M2  =  932  s.  2d.;  932  s.-f-20==:46  sov 
12  s.  Ans.   46  sov.  12  s.  2  d 

(IStJ,  page  153.) 
Ex.3.  5lb.  X12  +  7  oz.=:67  oz. ;  67  oz.x  20  +  12  pwt.^ 
1352  pwt.;  1352  pwt.  x  24  +  9  gT.=: 32457  gr.,  Aiis 
Ex.4   43457  gr.-^24=rl810  pwt.  17  gr.  ;  1810  pwt.^ 
20  =  90  oz.  10  pwt.;  90  oz. +-12  =  7  lb.  6  oz. 
Ans,  7  lb.  6  oz.  10  pwt.  17  gr. 
Ex.  5.  41760  gr. -^24  =  1740  pwt.  ;l740pwt.-^20=::87oz.; 

87  oz.-r-12i=7  lb. +  3  oz.,  Ans, 
Ex.  6.  14  lb.  10  oz.  18  pwt.  =  3578  pwt. ; 

3578  X  ^$.75=$2683.50,  Ans, 
Ex.  7.  5  lb.  6  oz.=1320  pwt. ;  2  oz.  15  pwt.=:55  pwt. ; 

1320-^55  =  24,  Ans, 
Ex.  8.  1  lb.  1  pwt.  16  gr.=5800  gr. ;  4  pwt.  20  gr.=  116  gr. ; 
5800-M16=:50  ; 
$1.25  X  50  ==$62.50,  Ans, 

(I865  page  155.) 
Ex.3.  16  lb.  X  12 +  11  oz.  =  203  oz. ;  203  oz.  x  8  +  7  dr. 

=  1631  dr.;  1631  dr.x3  +  2  sc.  =4895  sc; 

4895  sex  20+19  gr.  =  97919  gr.,  Ans, 
Ex.4.  471b.  X  12  +  6  |  =570  3  ;  570  |  x8  +  4  3 

=  4564  3  ;  4564  3  x  3  =  13692  3,  Ans. 
Ex.  5.  20  gr.x  5x365  =  36500  gr. ; 

36500  gr.-^20  =  1825^;  1825  3-^-3  =  608  31  3; 

608  3  +-8  =  76  5  ;  76  |  -M2  =  6  lb.  4  ^  . 
Ans,  6  lb.  4  3  13. 

(187 5  page  156.) 

Ex.3.  3  T.X20  +  14  cwt.=74  cwt. ;  74  cwt.xlOO  +  74 
lb.=7474  lb.;  7474  lb.  x  16  +  12  oz.=  11959e 
oz.;  119596  oz.x  16  + 15  dr.=  1913551  dr.,  Ans. 


68  COMPOUND    NUMBERS. 

Ex.  4.  1913551  dr.~16  =  119596  oz.  15  dr.;  11959G  02. 
-M6  =  7474  lb.  12  oz. ;  7474  lb. -^100=74  cwt. 
74  lb.;  74  cwt.-^20=:3  T.  14  cwt. 

Ans,  3  T.  14  cwt.  74  lb.  12  oz.  15  dr. 

Ex  5.     3  T.  15  cwt.  20  ]b.=:7520  lb. 

Ans.  $.22  X  7520=:$1654.40. 

Ex.  6,     115  lb.-=-2000=.0575  T.  ;  $10  x  .0575=i$.575, 

Am 

Ex.  7 
217  lb.  X  10  =  2170  lb.  @  $.06  =$130.20 

306  1b.  X    5=:1530  lb.  @  $.07A  =   114.75 


3700  lb.  @  $.08   =$296.00,  which  — $244.95 
=$51.05,  Ans. 

Ex.  8.     2  T.  X  2000  =  4000  lb. ;  4000  x  $.12i-=$500  ; 
$500  — $360  =  $140,  Ans. 

Ex.  9.     10  T.  X20  +  6  cwt.  =  206  cwt.;  206  cwt.  x  4  +  3  qr 
=  827  qr. ;  827  qr.x  28  +  14  lb.=23l70  lb. 

$.06    buying  price. 
$130-r-2000  =  .065  selling  price. 

$.005  gain  per  pound. 
$.005x23l70  =  $115.85,  Ans. 

Ex.  10.  2352  lb.-v-56=42  bu.; 

$.90  X  42  X  2  =  $75.60,  Ans. 

Ex.  11.  300  bbl.  X  196  =  58800  lb.,  Ans. 

Ex.  12.  $1.25  X  3  =  $3.75  cost^ 

$.0075x280x3=  6.30 

$2.55,  Ans. 


REDUCTION.  59 

(191,  page  157.) 
Ex.  1.     6  lb.  10  oz.=90  oz. ;  $.50  x  90  =  $45.0C   .ost. 

$J2_x_8_><_437^ x^^^^8.Y5  sold  lor. 

480  

$33. 75,  Ans. 

Ex.  2.     424  dr. -^8  =53  oz.  ;  53  oz.-t-12  =  4  lb.  5  oz.,  Jps 

Ex.  3.     20  lb.  8  oz.  12  pwt.  =  119328  gr. 

119328  gr.^7000=:l7gV3  lb.,  Ans. 
Ex.4.     $.40x16x20  =$128  cost 

$.50x320x437.5 


480 


.=1   145.83^ 


$  17.831,  Ana, 


(193,  page  159.) 

Ex.  3.  7912  mi.  x  63360  =  501304320  ia.,  Ans. 

Ex.4.  168i74  ft.-^3  =  56158  yd.;  56158  yd. -^5^  =  10210 
id.  3  yd.;  10210  rd. -4-40  =  255  fur.  10  rd. ;  255  fur.-^-8  =  31 
mi.  7  fur.  Ans.   31  mi.  7  fur.  10  rd.  3  yd. 

Ex.5.  31  mi.x84-7  fur.=255  fur.;  255  fur.x40-fl0 
rd.=10210  rd.;  10210  rd.  x  5i  +  3  yd.=56158  yd.;  56158 
yd.  X  3  =  168474  ft.,  Ans. 

Ex.6.  2500  fathoms  X  6  =  15000  ft.;  15000  ft. -^161  = 
909  rd.  l|ft. ;  909  rd.-4-40  =  22  fur.  29  rd. ;  22  fur.-f-8  = 
2  mi.  6  fur.  Ans.   2  mi.  6  fur.  29  rd.  1|  ft. 

Ex.  7.  2200  mi.  X  5280  =  11616000  ft.; 

$.10  X  11616000  =  $1161600,  Ans. 

Ex.8.  4  fathoms  X  6 +  3  fl.=27  ft.;  27  ft.  xl2+8in.= 
832  in  Ans. 

Ex.  9   200  mi.  =  12672000  in. ;  18  ft  4  in  =  220  in. ; 

12672000-4-220  =  57600  times,  Ans. 
Ex.  10.  120  lea.  x  3  =  360  geo.  mi ;  360  geo.  mi  x  1.15 - 
4  14  Eng.  mi.,  Ans, 

Ex.  11.  141  hands  x  4=58  in.,  Ans. 


60  COMPOUND   NUMBERS. 

(1045     page   160.) 

Ex.1.  3  mi.  X  80 +  51  d\.^291  cb. ;  201  ckxlOO-i- 
1S  l.  =  291'73  1.,  Ans. 

Ex,  2.  291'73  l.-^100--291  ch.  73  1. ;  291  ch,-v-80  =  3  ml 
51  ch.  Ans.  3  mi.  51  ch.  13  L 

Ex  3.     17  cL  31  1.  =  17.31  ch. 
12  ch.  87  l.  =  12.87  ch. 

30.18  ch.  half  romid  the  field. 
30.18  ch.  X  2  X  66  =  3983.76  ft.,  Ans, 

(lOGj  page  163.) 

Ex.3.  87  A.X4  +  2  R.  =  350  R. ;  350  R.  x  40 -f-38  sq.  rd 
=  14038  sq.  id. ;  14038  sq.  rd.  x  30^  +  7  sq.  yd.==424656i  sq 
yd.;  4246561  sq.  yd.  x  9  +  1  sq.  ft.  =  3821909i  sq.  ft.; 
38219091  sq.  ft.  X  144  +  100  sq.  in=:550355068  sq.  in.,  Ans, 

Ex.  4.  550355068  sq.  in. +-144  =  3821910  sq.ft.  28sq.in.; 
3821910  sq.  ft. -+9  =  424656  sq.  yd.  6  sq.  ft. ;  424656  sq.  yd.-+ 
301  =  14038  sq.  rd.  61  sq.  yd.;  14038  sq.  rd.+-40  =  350  R. 
38  sq.  rd.;  350  R.+-4  =  87  A.  2  R. 

Ans,   87  A.  2  R.  38  sq.  rd.  6i  sq.  yd.  6  sq.  ft.  28  sq.  in. 

But  (-J-  sq.  rd.)=:4  sq.  ft.  72  sq.  in. 

Hence,  Ans,  87  A.  2  R.  38  sq.  rd.  7  sq.  yd.  1  sq.  ft.  100  sq  in. 

Ex.  5.  100  X  30  =  3000  sq.  rd.=:18  A.  3  R.,  Ans, 

Ex.  6.  4  mi.  x  320  =  1280  rd.,  Ans, 
Ex.  7.  2  mi.  x  320  =  640  rd.,  Ans, 
Ex.  8.  100000  sq.  ft.  +-  9  =11111  sq.  yd.  1  sq.  ft.; 
IIIU  sq.  yd.+-30i  =  367  sq.  rd.  91  sq.  yd 
367  sq.  rd.+-40  =  9  R.  7  sq.  rd. 
9  R.+-4  =  2  A.  1  R. 
Ans.   2  A.  1  R.  7  sq.  rd.  9i  sq.  yd.  1  sq.  ft. ;  or 
2  A  1  R.  7  sq.  rd.  9  sq.  yd.  3i  sq.  ft. 


REDUCTION.  61 

Ex.  9.     181x16  =  296  sq.ft.; 

296  sq.  ft.-f-9=:32f  sq.  yd.,  Ans. 

Ex.  10.  (18  4- 161)  X  2  =  69  ft.,  distance  round  the  room ; 

69x9 

-69  sq.  yd.,  in  tlie  walls; 


9 
18x161 


=  83  sq.  yd.,  in  ceiling; 


69  sq.  yd. +  33  sq.  yd.  =  102  sq.  yd. 
$.22xl02=:$22.44,  Ans, 

Ex.  11.  40x20x2  =  1600  sq.  ft.  =  16  squares; 
$10xl6=$160,  Ans. 

(lOr,  page  164.) 
Ex.  2.     3686400  P.-M02400  =  36  sq.  mi.,  Ans. 

Ex.  3.     94  A.  X  10  +  7  sq.  ch.  =  94'7  sq.  ch. ; 
947  sq.  ch.xl6  +  12  P.=  15164  P.; 
15164  P.   X  625  4-118  sq.  1  =  9477618  sq.  l,  A^^s. 

Ex.  4.     4550000  sq.  l.-M0000  =  455  A. 
$50x455=$22750,  Ans, 

(lOO,  page  166.) 

Ex.  1.    125  cu.  ft.  X  1728  +  840  cu.  in.  =  216840  cu.  in.,  ^ai5. 
Ex.2.    5224  cu.  ft.-M28=40if  Cd.,  ^715. 
Ex.  3.    3ft.2in.=38in.;  2ft.  2  in.=26  in.;  1  ft. 8  in. =20 in.; 
88  X  26  X  20  =  19760  cu.  in.,  Ans. 

Ex.  4.     6x6x6  =  216  cu.  ft.; 

216  cu.  ft.  X  1728  =  373248  cu.  in.,  Ans. 

Ex.  5.  60  X  20  X  15  =  18000  cu.  ft. ; 

18000  cu.  ft.-^128  =  140|  Cd.,  Ans. 

Ex.  6.  10  X  31  X  3i  =  113f  cu.  ft.,  Ans, 
Ex.  7.  128^(3  X  12)  =  3f  ft.  high,  Ans 


62  COMPOUND    NUMBERS. 

Ex.  8.     21x115  lb.=4725  lb.=2  T.  1  cwt.  25  lb.,  Ann 
Ex.  9.     32  ft. +  24  ft.=i56  ft.;  56  ft.  x  2=^112  ft.  girt; 
112  X  1|  X  6  =  1008  cu.  ft.;  1008  cu.  ft.-^24^—A0j\   Pch. 
$1.25  X  40y\=:$50.909  -f ,  Arts. 

32  X  24  X  6 
Ex.10.    $.15  X — =$25.60,^715. 

Ex.  11.     10x9x8  =  720  cu.  ft.; 

'720+-10  =  Y2  minutes,  Ana. 

Ei.  12.     30  X  20  X  10=6000  cu.  ft. ; 

6000        ,,      .     , 

-=12  minutes,  Arts, 

50x10  ' 


(SOO,  page  168.) 

Ex.  3.  3  hhd.  X  2016  =  6048  gi.,  A7is, 

Ex.  4.  6048  gi.+2016  =  3  hhd.,  Ans. 

Ex.5.  13  hhd.  X  63  +  15  ga].  =  834  gal.;  834  gal.  x  4  x  1 
qt.  =  3337  qt. ;  3337  qt.  x  2  =  6674  pt.,  Ans. 

Ex.  6.  6674  pt.-^2  =  3337  qt. ;  3337  qt.-v-4  =  834  gal.  1 
qt.;  834  gal.-v-63  =  13  hhd.  15  gal. 

13  hhd.  15  gal.  1  qt.  Ans. 

Ex.  7.  1  hhd.=2016  gi. ;  $.06  x  2016=$120.96,  Ans, 

Ex.  8.  $2  X  10  =  $20  cost ;  $.05x  4  x  31^  x  10=$63  reed. 
$63  — $20  =  $43  gain,  Ans. 

Ex.  9.     $3.84-T-$.06  =  64  pt.=8  gal.,  Ans. 
Ex.  10.     2  gal.  2  qt.  1  pt.=21  pt.;  1  hhd.=504  pt. ; 
604-4-21  =  24,  Ans. 

(SOI,  page  169.) 

Ex.1.     49  bu.  x4  +  3  pk.=199  pk. ;  199  pk.  x8  +  7  qt,= 
1599  qt. ;  1599  qt.  X  2  + 1  pt.=3199  pt.,  Ans. 


REDUCTION.  68 

Kx.  2.  3199  pt.-^23:=1599  qt.  1  pt. ;  1599  qt.-^8  =  l99  pk. 
7  qt;  199  pk.^4r=:49  bu.  3  pk. 

Ans.   49  bu.  3  pk.  7  qt.  1  pt. 

Ex.  3.    1  bu.  x4  +  l  pk.=5  peck;  5pk.  xS  +  l  qt.— 41  qt, ; 
41  qt.x2  +  l  pt.=83  pt.,  Ans. 

Ex.  4.     83  pt.-^2rr41  qt.  1  pt.;  41  qt.-f-8=:5  pk.  1  qt. 
5  pk.-^4  =  l  bu.  1  pk.  Ans,  1  bu.  1  pk.  1  qt.  1  pt. 

Ex.  5.  $.Q5  X  60=$32.50  cost ; 

$.25  X  4  X  50  =  $50.00  sold  for ; 

$17.60  Ans. 
(205,  page  170.) 

Ex.  1.     1  bu.  (Dry  Measure) =21501  cu.  in.; 

2150|  cu.  in.-^57j  =  37;^J  wine  quarts; 
37±|  qts.— 32:=5i|  qts.,  Ans, 

Ex.  2.     40  qt.-f-4=rl0  gal. ;   10  gal.  x  282  =  2820  cu.  in.  , 
2820  cu.  in.-^57f  =  48||  qts.  Wine  Measure : 
48^  qts.~40  qts.  =  8f|  qts.,  Ans, 

Ex.  3.       1  bu.  Dry  Measure  =2150f  cu.  in. 
32  qt.  Wine  Measures  1848    cu.  in. 

302|  cu.  in.,  Ans. 

(206,  page  171.) 

Ex.  1.  365  da.  X  24  4-5  h.=8765  b. ;  8765  h.  x60f48 
min.=525948  mi»;t  625948  min.x  60+46  se  us=s 
31556926  sec,  Ans, 

Ex.  2      31556926  sec. -^60  =  525948  min.  46  sec; 

525948  min-v-60  =  8765  h.  48  min.;  8765  h.-^24 
=365  da.  5  b.      Ans.  365  da.  5  b.  48  min.  46  sea 


64  COMPOUND    NUMBERS. 

Ex.3.  5  wk.  x7  +  l  da.  =  36  da.;  36  da.  x  24+1  h,= 
865  h. ;  865  li.  x60  +  l  niin.:=51901  mm.;  51901 
min.  x60  +  l  scc.  =  31 14061  sec,  Ans, 

Ex.  4.  3114061  sec. -^60  =  51901  mm.  1  sec;  51901  mm. 
-^60:=865  h.  1  min.;  865  h.-^24  =  36  da.  1  h. ; 
36  da.-^-7  =  5  \vk.  1  da. 

Ans,  5  wk.  1  da.  1  h.  1  min.  1  sec 

Ex.  6.     10  mi.=:l'7600  yd.; 

17600  sec-^60==293  min.  20  sec  ;  293  min.-^60 
=4  h.  53  min.  Ans,  4  h.  53  min.  20  sec 

Ex.7.     29  da.x24+12  li.=708  h. ;  708  h.  x  60  +  44  min, 
=42524  min. ;  42524  min.  x  60  +  3  sec  = 
2551443  sec,  Ans, 

Ex.8.     40    yr.x365^z=:14610    da.;     14610    da.  x45  = 
657450  min.  gained. 

657450  min.-f-60=rl0957  h.  30  min.;  10957  h.-^ 
24=456  da.  13  li.      Ans,  456  da.  13  h.  30  min. 

(207,  page  173.) 

Ex.1.     10  S.  x30  +  10°=310°;  310°  x  60  +  10'=18610' ; 

18610'x60  +  10''=1116610^  Ans, 
Ex.  2.  1116610'' -^60  =  18610'  10'';  18610'-^60  =  310°  10'; 

310°-^30=:10  S.  10°.       Ans.  10  S.  10°  10'  10". 
Ex.  3.     11400'-^60  =  190°,  Ans, 
Ex.  4.     190°  X  691  =  13148  miles,  Ans. 
Ex.  5.     360°  X  60  =  21600',  Ans, 
Ex.  6.     397'-^60=6°  37',  Ans, 

(210,  page  174.) 

Ex.1.     150000000-^12  =  12500000  doz.; 

12500000  doz. -^12  =  1041666  gross  8  doz.; 
1041666  gross-^12  =  86805  great  gross+6  gross, 
Ans,  86805  great  gross  6  gross  8  doz. 


REDUCTION.  65 

Ex.  2.     100000  sheets-^24r=4166  quires  16  sheets; 
4166  quires-^20  =  208  reams  6  quires; 
208  reams-^2r=104  bundles ; 
104  bundles-4-5  — 20  bales  4  bundles. 

Ans.  20  bales  4  bundles  6  quires  16  sheets. 

Ex,  3.     20  years  x  4  + 10  years =90  years,  Ans. 

Et   4.     8  sheets  x  8=64  leaves  ;  64  leaves  x  2  =  128  pages, 

Ans, 
Ex.  5.     32  pages xl0x2  =  640  pages,  Ans, 

PROMISCUOUS    EXAMPLES    IN    REDUCTION. 

Ex.  1.     6  yd.  3f  qr.  =  27J  qr.;  333  yd.  =  1332  qr. ; 

1332-^273  =48  suits,  Ans, 
Ex.  2.     1  oz.  15  pwt.i=:35  pwt. ; 

I.YOx  35  =$24.50,  Arts. 
Ex.  3.     2  lb.  3  3   5  3   13  10  gr.=13290  gr.; 

13290-^15  =  886,  Ans, 
Ex.  4.     1  T.  11  cwt.  12  lb.  =  3112  lb.; 

3112x$.01i=$38.90,  Ans, 
Ex.  5.     1456  lb.-^32  =  45.5  bu. ; 

$.375  X  45.5=$17.0625,  Aiis. 
Ex.  6.     45  lb.  X  1000  =  45000  lb. 

45000  lb.-M96  =  229  bbl.  116  lb.,  Ans. 
Ex.  1,     2430  lb.-T-60c±40.5  bu.  ; 

$1.20  X  40.5=$48.60,  Ans, 
Ex.  8.  $12.50-^200=$.06i,  Ans, 
Ex.  9     360°  X  69.15=24897.6  stat.  mi.  ; 

24897.6  X  63360=1577511936,  Ans. 
Ex.  10.  10  mi.  X  80 +  7  ch.  +  l  eh. (4  rd.)=808  ch. ;  808 
nh.  X 100  +  20  1.= 80820  L,  Ans. 


1j6  compound  numbers. 

Ex.  11.     25  X  100  X  144  =  360000  sq.  in. ; 

$.01  X  360000  =  13600,  Ans. 
Ex.  12.     50  X  25  X  10  =  12500  cu.  ft. ; 

12500  cu.  ft. -^1 6  =  781  cd.  ft.  4  cu.  ft.; 

'781  cd.  ft.^8=97  Cd.  5  cd  ft. ; 

Ans,  97  Cd.  5  cd.  ft.  4  cu.  ft 

Ex  13,     10  X  10  X  10  X  1728  =  1728000  cu.  in. ; 

1728000  cu.  in.-v-231  =  74804^  gal.; 

1i80^  gal.-^63  =  118   hhd.  464-f  gal.,  ^W5. 
Ex.  14.     8  X  5  X  41=180  cu.  ft.=311040  cu.  in. ; 

311040  cu.  in.-T-2150.4  =  144y\  bu.,  Ans. 

Ex.  15.  Mar.  31  da.  June  30  da.  Sept.  30  da, 
Apr.  30  da.  July  31  da.  Oct.  31  da. 
May  31  da.         Aug.  31  da.         Nov.  30  da. 


Spring,  92  da.  Summer,  92  da.  Autumn  91  da. 

92  da.— 91  da.  =  l  da.  =  86400  sec,  Aiis. 

Ex.  16.     1296000  sec.-=- 86400  =  15  da.,  Ans. 

^     ,^      20x13      ,^     ,      . 
Ex.  17. =40  yd.,  Am. 

Ex  18.     4  reams  X  20  +  10  quires= 90  quires;  90  quires  x 
24  +  10  sheets=2l70  sheets,  Ans. 

Ex.19.     16  ft.  6  in.=l  rd. ;  1  mi.  =  320  rd. ;  320^1  = 
Ifto  times  in  1  mi.  320  x  42  =  13440  times,  Ans. 

)|Ex.  20.       1000000  sec.-^60  =  16666  min.  40  sec. ; 
16666  min.-f-60  =  277  h.  46  min. ; 
277  h.-f-10  =  27  da.  7  h. ; 
Ans.  27  da.  7  h.  46  min.  40  sec. 

Ex.  21.     6x4^  =27  sq.  mi ;  — — — =216  farms,  Ans. 

Ex.  22.     10  mi.  176  rd.=3376  rd. 

$21.75  x  3376  ^$73428,  Ans. 


REDUCTION.  til 

(21 1 5  pagelTG.) 
Ex.  2.     j-oV  f  iJ  X  V  X  -V  =  2«5  d.,  A71S, 

Ex.  3.       y^  ^  0  0   Wk.  X  I  X  V-  X  -¥-  =  f  0   ^^^-j  ^'^*- 

Ex.  4.  y-,V2  ^^^^'  X  r-  X  f  X  f-  X  f - 1  gi.,  ^ws. 

Ex  5.  ^J-o  oz.  X  -/  X  V  — i  gr.,  J.71S. 

Rv   fi  1 mi  X  ^  X  ^-"  X  -3-3-  X  J^=  J-9-®-,  in..  Ans 

Ex.  7.  f  X  i  X  f  lb.  X  Y  =1  oz-j  ^^^5« 

Ex.  8.  elo  hhd.  X  V-  X  f  X  2.  =  f  J  pt.,  ^?^5. 

Ex.  9.     yyVo  A.  X  f  X  V-=i-  rd.,  ^n5. 

(SIS,  page  177.) 

Ex.  2.     i  ft.  X  3=3  =  ri2  r^-  ■^^^^• 
Ex.  3.     f  dr.  X  j\  X  tV^tAo  lb.,  ^7^5. 
Ex.  4.     J  ct.  X  y^Vo  =  2  oVo  E.,  ^?^5. 
Ex.  5.     i  ft.  X  3  J-^=y^i^^  mi.,  Ans. 
Ex.  6.     -f  X  t  pwt.  X  ^V  X  T2  =  2 « 0  lt>.,  Ans. 
Ex.7,     f  pt.x-Lxix^V=8io  l^bd. 

^  hhd.—- gJ  0  liiid.=:f  J-f  hhd.,  Ans. 
Ex.  8.     I  in.  X  c  3 k 0  =To  jVo  0  i^i-,  ^'^s- 
Ex.  9.     f  oz.  X  tV=:2\  lb— 2T  <^^'  2  lb. ;  and  ^V  of  2  lb.  is 

I  of  f  of  2V  of  2  lb.,  or  I  of  ^  of  2  lb.,  Ans. 
Ex.  10.  f  oz.  X  tV=2"t  lb.  =  2T  of  2  lb. ;  and  ^\  of  2  lb.  ia 

J.  of  f  of  J,  of  2  lb.,  or  I  of  f  of  2  lb.,  Ans. 

(2185  page  178.) 

Ex.2.     4mo.x30  =  l7|  da.;  -Ida.  x24  =  3f   h. ;  ^  h.  x 
60  =  254  rain. ;  4  min.  x  00  =  424  sec. 

A71S.  17  da.  3  b.  25  min.  42^  sec. 

Ex.  3      |£  X  20  =  84  s. ;  4  s.  X  12  =  64  d. ;  4  d.  x  4  =  34  ^ 

A71S.  8  s.  6  d.  34  far. 


68  COMPOUND  NUMBERS. 

Ex.  4.  §  bu.  x4  =  lj  pk. ;  J  pk.x8  =  4f  qt.;  f  qt  x2-= 

1|  pt.  Ans,  1  pk.  4  qt.  If  pt. 

Ex.5.  4  of  15  cwt.  =  124  cwt. ;  f  cwt.  xl00=:85f  lb.; 
f  lb.  X  16  =  llf  oz. ;  f  oz.  x  16  =  6^  dr. 

Ans.  12  cwt.  85  lb.  11  oz.  Of  dr. 
Ex.6.  |x-}x  V^3=4|f  oz. ;  |^  oz.  x  IGnzllf^  di. 

Ans.  4  oz.  11|^  dr. 
Ex.  7.  I  A.  X  4  =  3i  R. ;  ^  R.  x  40  =  131  P. 

^m\  3  R.  131  p. 
Ex.  8.     If  da.  X  24r=16/3  h. ;  j\  b.  x  60  =  36||  min. ;  \i 
inin.  X  60  =  55  j^^  sec. 

Ans,  16  b.  36  min.  55-^^^  sec. 
Ex.  9.     I  lb.  X  12  =  '7i  oz. ;  i  oz.  x  20  =  4  pwt. 

Ans,  7  oz.  4  pwt, 
Ex.  10.  i  ofVT.=4/_  T. ;  /^  T.x20=5f  cwt;  f  cwt 

X  100  =  55f  lb.  Ans.  4  T.  5  cwt  55|  lb. 

Ex.  11.  I  ofY  A.  =  lf  A.;  f  A.  X  4  =  11  R. ;  i  Rx40  = 
20  P.  Ans.  1  A.  3  R.  20  P. 

(214,  page  179.) 

Ex.  2.     6  fur.  26  rd.  3  yd.  2  ft=4400  ft ;  1  mi.=5280  ft; 

UU  mi.=f  mi.,  Ans. 
Ex.  3.     13  s.  7  d.  3  far.=655  far. ;   l£  =  960  far.  ; 

960    ^  —  192    "^i   -^rx/Cis, 

Ex.  4.     10  oz.  10  pwt  10  gr.  =  5050  gr.;  1  lb.  =  5760  gr.; 

HU  Ib.-Hf  lb.,  Ans. 
Ex.  5.     2  cd.  ft  8  cu.  ft =40  cu.  ft ;  1  Cd.  =  128  cu.  ft ; 

tVV  Cd.=tV  Cd.,  Ans. 
Ex.6.  lbbl.lgal.lqtlptlgi.  =  1053gi.;  lLbd.=2016gi»^ 

i^fl  bbd.=iif  hbd.,  Ans. 
Ex.  7.     4  yd.  li  ft  =  27  half-feet;  2  rd.=66  half-feet; 


REDUCTION.  89 

Ex.  8.     -p  bu- =—-=:-  bu.,  Ans. 
4  20     5 

Ex.  9.     -^j  =  ij  Ans. 

Ex.  10.  2  yd.  2  qr.=10  qr ;  8  yd.  3  qr.=:35  qr. ; 
H  yd.==f  yd.,  Ans. 

(215>5  page  180.) 

Ex.2.     .217°  X  60=13.02';  .02'x  60  =  1.2^ 

Ans,  13'  1.2^ 

Ex.3.     .659wk.  X  7=4.613  da. ;  .613  da.  x  24  =  14.712  h.; 
•712  h.  X  60  =  42.72  min. ;  .72  min.  x  60  =  43.2  sec. 
Ans,  4  da.  14  h.  42  min.  43.2  sec. 

Kx.  4.     .578125  bu.  X  4  =  2.3125  pk.;  .3125  pk.  x  8=2.5  qt.; 
.5  qt.  X  2  =  1  pt.  Ans,  2  pk.  2  qt.  1  pt. 

Ex.  5.     .125  bbl.  X  31.5  =  3.9375  gal. ;  .9375  gal.  x  4  = 
3.75  qt. ;  .75  qt.  x  2  =  1.5  pt. ;  .5  pt.  x  4  =  2  gi. 
Ans.  3  gal.  3  qt.  1  pt.  2  gi. 

Ex.  6.     .628125  £  x  20  =  12.5625  s. ;  .5625  s.  x  12  =  6.75  d.; 
.75  d.  X  4  =  3  far.  A7is,  12  s.  6  d.  3  far. 

Ex.  7.     .22  bhd.  x  63  =  13.86  gal. ;  .86  gal.  x  4  =  3.44  qt. , 
.44  qt.  X  2  =  .88  pt. ;  .88  pt.  x  4  =  3.52  gi. 

Ans,  13  gal.  3  qt.  3.52  gi. 

Ex.   8.     .67x3.45  =  2.3115;  .3115x320  =  99.68; 
.68x16^=11.22;  .22x12  =  2.64. 

Ans.  2  rai.  99  rd.  11  ft.  2.64  in. 

Ex.  9.  .42857  mo.  x  30  =  12.8571  da.;  .8571  da.  x  24  = 
20.5704  h. ;  .5704  h.  x  60  =  34.224  min. ; 
.224  min.  x  60^13.44  sec. 

Ans.  12  da.  20  h.  34  min.  13^}  sec. 


Te 


COMPOUND    NUMBERS. 


Ex.  10.  .78875  T.  x  20  =  15.775  cwt.;  ,775  cwt.  x  4=3.1  qr.; 
.1  qr.  X  28  =  2.8  lb. ;  .8  lb.  x  16  =  12.8  oz. 

Ans.  15  cwt.  3  qr.  2  lb.  12.8  oz. 

Ex.  11.  .88125  A.  X  4  =  3.525  R. ;  .525  E.x  40  =  21  P.; 

Ans,  5  A.  3  R,  21  P. 

Ex    12.  .0055  T.x  2000  =  11  lb.,  Ans. 

Ex.  13.  .034375  bundles x 40  =  1.375  quires;  .-^>75  quires x 
24=9  sheets;  Ans,  1  quiie  9  sheets. 

(916,  page  181.) 


Ex. 


1.00  gi. 


1.250  pt. 


3.625  qt. 


Ans.     .90625  cral. 


Ex.  3. 


24 
20 
12 


9.000  gr. 


13.375  pwt. 


10.66875  oz. 


Ans.     .8890625  ib 


Ex.4. 


2 

4 

1.2  pt. 
.6  qt. 

63 

.150  gal. 

Ans.     .00238 +  hhd. 


Ex.  5. 


1.12  qt. 


3.14  pk. 


Ans.     .785  bu. 


Ex.  I 


40 
4 


12.56  P. 


3.314  R. 


Ans.     .8285  A. 


Ex.  7.    12 

3 

5,5 

40 

8 


6  in. 


1.5  ft. 
iT^yd. 


3.1818181+  rd. 


.07954545  + fur 


Ans.     .00994318  + mi. 


Ex.  8. 
Ex.  9. 
Ex.  10. 
Ex.  11. 
Ex.  12. 


.32  pt.-h64=.005  bu.,  Ans. 
4.875  ft.+6  =  .8125  fathoms,  Ans. 
150  sheets-^480  =  .3125  Rm.,  Ans. 
47.04  lb.-f-196=.24  bbl  flour.,  An^ 
.33  ft.-^5280  =  .0000625  mi.,  Ans, 


ADDITION. 


n 


Ex.  13. 


60 

51.6  sec. 

60 

36.96  min. 

24 

5.616  h. 

Ans.     .234  da. 

ADDITION. 

(Sir,  page  183.) 

Ea    3.     43  10.  1    3   2  ^  16  gr.,  Ans. 
Ex.  5.     68  bu.  3  pk.  1  qt.  1  pt.,  Ans, 
Ex.  6.     21  mi.  5  fur.  23  rd.  1  yd.  4  in.,  Ans. 
Ex.  10.  627  hbd.  1  gal.  1  qt.  1  pt.,  Ans. 
Ex.  11.  187  bu.  3  pk.  1  pt.,  Ans. 

Ex.  16.  152  cu.  yd.  9  cu.  ft.=:152i-  cu.  yd. ) 
$.16xl52J-  ===$24,371.         ) 

Ex.  17.  2564  lbs. 
2713    " 
3000   " 
3109    " 

~"  =203.3214+  bu. 

2.657  +  ,  At^. 


Ex. 


An3» 


11386  lb«i,- 

-56 

=: 

$.80  X 

203.3214: 

bbls.  gal. 

qt. 

pt. 

gi. 

18.   1      4 

0 

1 

0 

30 

2 

0 

1 

2   15 

0 

0 

0 

3  49     2     1      1=:4613  gi. 
$.09x4613=:$415.l7,  Ans. 


(21 85  page  185.) 

Ex.  2      i  rd.  =  12  ft.  4i  in. 

I  ft.  = 9_  " 

13  ft.  11  in.,  Ans. 


T2  COMPOUND  NUxMBERS. 

Ex,  3.     1  mi.  =  7  fur. 

I  fur.=  26  rd.  11   ft. 

I  rd.  =  13    "    9  in. 


7  fur.  27  rd.    1^  ft.  9  in.  ;  or 
1  fur.  27  rd.    8    ft.  3  in.,  Am. 


Ex.4.     |£=13s.     4  d. 


^  s.  =      6  "  2|  far. 

13  s.  10  d.  2|  far.,  A7is, 

Ex.  6. 

f  T.  =12  cwt. 

^  cwt.=      42  lb.  134  oz. 

12  cwt.  42  lb.  13f  oz.,  An4. 

Ex.  6. 

f  da.  =  9  h. 

i  h.  =    30  min. 

9  h.  30  min.,  Ans, 

Ex.  7.     1  wk.  =  l  da.    4  h. 
f  da.  =  18  " 

^  h.    =  15  min. 

1  da.  22  h.  15  min.,  Arts, 

Ex.  8.     -f  lihd.  =  54  gal. 

f  gal.  = ^qt. 

64  gal.  3  qt.,  Ans. 

Ex.  9.       4    cwt. =64  lb. 

8f    lb.    =   8  "  13  oz.     5i  dr. 
3yV  oz.    =  3    "   14|   " 

73  lb.  1   oz.    3|i  dr.,  Ana. 

Ex.  10.   I  mi.  =  3  fur. 

|yd.=  2  ft. 

J  ft.  =  9  in. 

3  fur.  2  ft.  9  in.,  An8. 


SUBTRACTION.  T8 

Ex.  11.  1  of  }  A.=:i    A.  -  26  P.  181  ^  sq.  ft. 

60f  rd.  =  l  K.  20  "  204^^  « 
f  A.=l  "  5  "  194  If  " 
Y^  A.  ==  13    "     90  f      " 

3  K.  26  P.  126yV2  sq.  ft.,  Am, 

Ex.  12.     11    T.    =1  T.    3  cwt.  33  lb.    5^  oz. 
If^  T.    =1  "     3    "     75  " 
18f    cwt.=  18    "     83  "      6i  '•• 

3  T.    5  cwt.  91  lb.  10|  oz.,  Ans. 


SUBTRACTION. 

(2195  page  187.) 

Ex  4.     3  T.  18  cwt.  70f  lb.,  Ans. 
Ex.  6.     2953  £  2  s.  7f  d.,  Ans. 

Ex.  11.  365  da.  X  5  4-2  da.=1827  da. 
1  bhd.=63  gal. 
1827  gi.    =57     "    0  qt.  0  pt.  3  gi. 

6  gal.  3  qt.  1  pt.  1  gi.,  Ans. 

Ex.  12.  196  A.  2  R.  16.25  P. 

200   "    1   "   20         " 
177   "   0  "   36         " 


1st,  2d,  and  3d  own  574  A.  0  R.  32.25  P. 
640  " 


4th  owns     65  A.  3  R.    7.75  P.,  A71S. 


Ex.  13.  16  Cd.  5  cd.  ft.  75  Cd.  6  cd.  ft. 

24    "    6       "     12cii.ft.      69     "    2      "      12  cu.  ft. 

^^  * 6  Cd.  3  cd.  ft.    4  cu.  ft. 


69  Cd.  2  cd.ft.  12cu.ft.  Ans. 


K.  P. 


^  COMPOUND    NUMBERS. 

Ex.  14.  10  gal.  1  qt.  1  pt.  63  gal 

15    "  1  pt.  40    "    1  qt. 

14.     «     Q    «  

22  gal.  3  qt.,  Ans, 


40  gal.  1  qt.,  Ans. 

(32O5  page  189.) 

yr.      mo,    da.  yr.      mo.    da. 

Ex.  2.     1799  12   14  Ex.  5.     1815     G   18 

1732     2  22  1775     6  17 


67     9  22,  Ans.  40     0     1,  Ana 

jT.        mo.    da.    h.  min. 

Ex.  6.  1861  1  3  8  50 
1856  6  24  14  20 

4  6  8  18  30,  Ans, 

Ex.  7.  122  da. ;  244  da. ;  306  da.,  Ans. 

Ex.  8.     Erom  Nov.  6  to  April  6,  151  da. 
From  Apr.  6  to  Apr.  15,      9  " 

160  da.,  Am, 

Ex.  9.     From  Aug.  20  to  June  20,  304  da. 
Subtract       5    " 


299  da.,  Aru 
(221 5  page  190.) 


Ex.  2.     i  rd.=:8  ft.  3  in. 
|ft.  =  9   « 


7  ft.  6  in.,  Ans. 

Ex.  3.     f  £=11  s.  1  d.  li  far. 
4s.=  6" 

10  8.  7  d.  11  far.,  Aiu. 


MULTIPLICATION^.  75 

Ex.  4.     I  of  3.45  mi.=2.3  mi ; 
2.3  mi.— .7  mi.=:1.6  mi. 

1.6  mi.  =  l  mi.  4  fur.  32  rd.,  Ans. 

Ex.  5.     8yV  cwt.       =8  cwt.  3  qr.  16  lb.  12  oz.  12f  dr 
1  qr.  2^  lb.   =  1  qr.    2   "     6    "   13  4     " 

Ans.     8  cwt.  2  qr.  14  lb.     5  oz.  15/j  dr 

Ex.  6.     1  wk.=l  da.  9  h.  36  min. 
1  da.  =  4  "  48    " 

1  da.  4  li.  48  min.,  Ans. 

f  I  of  120  mi.=41  mi.  7  fur.  9  rd.  8  ft.  7^  in..  Am 

Ex.8.     l-i=J;  ioff=/^; 

y^  of  96  gal.=25  gal.  2  qt.  3i  gi.,  Ans 


MULTIPLICATION. 

(222,  page  192.) 

Ex.  4.    Ans.  23  £  13  s.  4  d. 

Ex.  5.  Ans.  23  lb.  4  oz.  6  pwt.  10  gr. 

Ex.  6.   ^7i5.  163  T.  1  cwt.  36  lb.  4  oz. 

Ex.  7.    ^W5.  128°  35'  15"^. 

Ex.  9.    Ans.  20fi)  1  |   3  3  1  3  16  gr. 

Ex.  10.   Ans.  235  mi.  6  far.  7  rd.  4^  ft. 

Ex.  13 


bu.     pk.    qt    pt. 

45     3      6      1 

8 

Ex.  14. 

1 

a. 

9 

d. 

12 

367     2     4 
8 

17 

16 

6 
10 

2941,  Ans  178       5,  ^w«. 


76  COMPOUND    NUMBERS. 

Ex.  15.     U800-T-$80  =  60  —  6  x  10. 

A.  R-  P.  8q.  yd.        8<i.  ft 

4  3  26  20  3 

6 

29  2  0  1 

10 

295  10,  Alls, 

Ex.  17.  Ans,  359°  45'40.45^         Ex.  18.    Arts,  6  libd. 


DIVISION. 


(223,  page  194.) 

Ex.  7.   Ans.  1  oz.  17  pwt.  4  gr. 

Ex.11.    5£  10  s.  10d.=1330d.; 

537£lOs.  10d.=129010  d. 
129010-^1330=^97,  Ans, 

^     ,^      50x30x6     ^^  .    ^         .^      . 

Ex.  12.     — =11  cu.  yd.  3  cu.  ft.,  Ans. 

2i  X  o  X  b 

^      ^^      6x5x640     ^^^^    . 
Ex.  13.      __  =  106|A. 

106  A.  2  R.  26  P.  20  sq.  yd.  1  sq.  ft.  72  sq.  in.,  Ans. 

Ex.  14.     4  bu.  3  pk.  2  qt.=154  qt. ; 

336  bu.  3  pk.  4  qt.  =  10780  qt. 
10780-^-154  =  70,  Ans, 

Ex    15.     60  yd.  2.25  qr.  =  242.25  qr.;  242.25  qr.x4  = 

969  qr. ;  |  of  969  qr.=646  qr. ;  9  yd.  2  qr.=:38  qr.  \ 
6464-38  =  17,  Ans, 


LONGITUDE   AND    TIME. 

LONGITUDE  AND  TIME. 
(325,  page  196.) 


Ex.  2.  84° 

24' 

Ex.  3.  155° 

74 

1 

18 

28' 

10° 

23' 

173° 

28' 

4 

,  Ans, 

4 

41  min. 

32  sec, 

11  li.  33  min.  52  sec.,  Ans. 

Ex.4.  ir 

1' 

Ex.  5.  118° 

+  122°=240°; 

SO 

19 

360° 

-240°  =  120°; 

107° 

20' 
4 

120°H 

rl5=:8  h.,  Ans. 

7  h.  9  min.  20  sec.,  Ans, 

Ex.  6.  12  h. 

77°  1'=  5  "     8  min.    4  sec. 


6  h.  51  min.  56  sec,  A.M.,  Ans, 
Ex.  7.     90°  15' 

63    36  4  li. 


26°  39'=  1  li.  46  min.  36  sec. 


2  h.  13  min.  24  sec,  P.M.,  Ans, 
Ex.  8.     124°— 67°:=57°  =  3  li.  48  min.,  Ans, 
Ex.  9.     99°    5 

68    47 


30°  18'=2  h,  1  min.  12  sec.  difference  of  time. 
Time  at  Bangor,  1859  yr.  1  mo.  1  da.  1  h.  0  min.    0  sec.  a.  m 
Subtract  2  h.  1  min.  12  sec. 


Time  at  Mexico,  1858  yr.  12  mo.  31  da.  10  h.  58  min.  48  sec.  p.m. 

Ans, 
Note.    In  the  above  subtraction,  borrow  31  days,  the  month  being 
December. 


78  COMPOUND    NUMBERS. 

(236,  page  197.) 

Ex.  2.     11  h.  33  min.  52  sec.  =  693  rain.  52  sec; 
(693  min.  52  sec.) -^4  =  173°  28',  Ans. 

Ex.  3.     7  h.  9  rain.  20  sec.=429  rain.  20  sec. ; 
(429  min.  20  sec.)-r-4  =  107°  20',  Ans. 

Ex.  4.     16  h.  30  rain.  at  St.  Petersburgh ; 

8  li.  32  rain,  36  sec.  at  New  Orleans ; 

7  h.  57  rain.  24  sec.=477  rain.  24  sec. 
(477  rain.  24  sec.)-T-4  =  119°  21',  Ans. 

Ex.  5.  74°  1'  West;      8  h.  40  rain.=130° 

4h.=60°  74°  r  West, 

1st  Ans,  14°  1'  West.  2d  Aris,  55°  59' East 

13  h.  25  rain.  =  201°  15' 

74°    I'West, 

3d  Ans.  127°  14'  East 


DUODECIMALS. 


Ex.  2 

13  ft   9' 
11' 

MULTIP 

(229, 

LIGATION. 

page  200 
Ex.  3. 

11  ft    9' 
1         3' 

12  ft.  7'  S 

r,  Ans. 

2       11' 
11          9 

3^^ 

14  ft     8' 

4 

68  ft     9', 

Ans. 

DUODECIMALS.  T9 


Ex.  4.     12  ft.  11'  6  ft.  2' 

4  2  ft.  4' 


51  ft.  8' length  of  walls;    2       0'     8' 
9  ft.  3'  12       4' 


12      11  14  ft.  4'     8" 

465  3 


477  ft.  11'  area  of  walls  ;  43  ft.  2',  windows  and  door; 
43  ft.    2' 


434  ft.    9'=48  sq.  yd.  2  sq.  ft.  9',  Arts. 

Ex.  5      30  ft.  4'  Ex.  6.     18  ft.  6' 

12  ft. 


25  ft. 

6' 

15 

2 

758 

4 

773  ft. 

6' 

12  ft. 

5' 

322 

3   6" 

9282 

9604  ft.  3'      6%  Ans, 

Ex.  7.     36  ft.  10'  Ex.  8. 

22  ft.    3' 


9 

2 

6 

810 

4 

819  ft. 

6' 

e'' 

5  ft. 

2' 

136 

7 

1 

4097 

8 

6 

222  ft. 

5  ft.  6' 

111 

1110 

1221  ft. 

1221  cu.  ft. 

-^- 

128  = 

9  Cd.  69  cu. 

ft. 

,  Ans, 

32  ft.  8 

9  ft. 

294  sq.  ft. 
294  sq.ft.-v-9  =  32|sq.yd 
$.l7x32|=$5.55i    Ans 


4234  ft.   3'     V  ; 

156  cu.  yd.  22  cu  ft.  3'  1\  Ans. 


80  COMPOUND    NUMBERS. 


Ex.  9.     33  ft.  9'  27'       3 


48  ft.  12x3     4 


aJ^' 


36 
1584  180-^f  =  240  yd,,  Ans. 


1620ft.  =  180  8q.  yd. 


DIVISION. 

(330,  page  201.) 

2.     16  ft.  8^)44  ft.  5'     4''(2  ft.  8',  Ans. 
33       4 


11       1      4 
11        1      4 


Ex.  3.     40  ft.  11'  4'')184  ft.  3'     0''(4  ft.  6',  Ana. 
163       9      4 


20       5      8 
20       5      8 


Ex.4.     2  ft.  r)14ft.  6'     (5  ft.  r  4''  +  ,  ^w». 
12      11 


1 

7      0^ 

1 

6      1 

11 

0'" 

10 

4 

8'",  rem. 


PROMISCUOUS   EXAMPLES.  81 

Ex.  5.  3  ft.    f         8  ft.  11'  6^)64  ft.  2'    b\l  ft.  2^  Arts 
2  it.    6'  62       8      6 


7        2 
1         9     6 

1 
1 

5    11 
5    11 

8  ft.  11'    Q" 

PROMISCUOUS  EXAMPLES. 

(Page  202.) 
Ex  1.     115200  g^.-^5760  =  20  lb.,  Ans, 

Ex.  3.     1560  bu.  X  4=6240  pk. ;  3  bu.  1  pk.=13  pk. , 

6240-^-13=480,  Ans. 

Ex.4.  295218  in.-^12  =  24601i  ft.;  24601i  ft.-r-16i=- 
1491  rd.;  1491  rd.-^40=37  fur.  11  rd. ; 
37  fur.~8=4  mi.  5  fur. 

Ans,  4  mi.  5  fur.  11  rd. 

Ex.  6.  3  X  20  X  24  =  1440,  Ans. 

^     ^      $3.25x4x20x6x10     ^  ^       , 

Ex.  7.     ^ =$121.87^,  Ans. 

Ex.  8.     1  bbl.=1008  gi. ;  1  qt.  1  gi.=9  gi.  ; 

1008^9  =  112,  Ans. 

Ex.  9.     $.6.40  X  Y  X  f =$980.10,  Ans. 

Ex.  11.  336  bu.  3  pk.  4  qt.= 10780  qt. ;  4  bu.  3  pk.  2  qt.= 
154  qt.;  10780-^-154=70,  Ans. 

Ex.  12.  3  qt.  1  pt.,  Ans. 

Ex.  13.  1  mi.=l760  yd.;  2  fur.  36  rd.  2  yd.=640  yd. ; 

j%f^^^  mi.=y\  mi.,  Ans. 


82  COMPOUJSTD    NUMBERS. 

Ex    (4.  2  da.= 172800  sec;  13  h.  26  min.  24  sec.= 

48384  sec. ;  rVVVVo  =  21,  ^^' 

Ex    1.5.  2G  A.  2  R.=106  R.;  5  A.  3  R.=:23  R., 
106  R.— 23  R.=83  R.;  j%\,  Ans. 

Ex.  16.  f£  =  12  s. 

51  d. 

Alls,  11  s.  6J  d. 


Ex.  17.         I  yd. =5|  in. 

^  ft.  =lf  " 
1  in.  =  4  ^^ 

7  in.,  ^715. 

Ex.  19.  1732  yr.  2  mo.  22  da. 
1706    "    1    "    18    " 


26  V-  1  rno.    4  da. 


Ex.  20.  87o  30 

71°     4'  12  h. 


16''  26'=  1  h.    5  min.  44  sec. 


10  h.  54  min.  16  sec,  A.  M.,  Aru^ 


Ex  22.     I  mi.  =5  fur.  13  rd.  5  ft.  6  in. 
^fur.=:  20   " 

rd.  =  2   "  9  " 


TJ 


5  fur.  33  rd.  8  ft.  3  in.,  Ans. 

Ex.  23    20  bu.  3  pk.  6  qt.=20.9375  bu. ; 

$.80  v:  20.9375=116.75,  A7is, 
Ex.  24.  .875  gross  x  12  =  10.5  =  10|  doz.,  Ans, 
Ex,  25.  56.5x24.6  =  1389.9  P.  =  8  A.  2  R.  29.9  P.,  Ans. 
Ex.  26.  20f  (lix2)  =  23  ft.,  length  of  one  side; 

23  X  8  X  11  X  4  =  1104  cu.  ft.,  by  the  mason's  rule  ; 

(see  note  5,  page  166). 

1104-^24.75  =  44.6+  Pch.,  Ans. 


PROMISCUOUS   EXAMPLES.  88 

El.  27,  640  bu.  3  pk.  produce  of  the  farm. 

160    "    0  **    6  qt.,  i  for  the  rent. 

480  bu.  2  pk.  2  qt.  to  be  shared  among  A^B,  and  G. 
170    "    2   "    6  "   A's  share. 


309  bu.  3  pk.  4  qt.  left  for  B  and  C. 
147    "    3  **    6  "   B's  share. 


161  bu.  3  pk.  6  qt.,  C's  share,  Ans, 

Ex,  28.  13  lb.  8  oz.  11.4  dr.  =  13.54453125  lb.  Troy. 

13.54453125  lb.  X  |Jf  =  16.46036783  +lb.  Av.= 
16  lb.  5  oz.  10  pwt.  11.7  4-gr.,  Arts. 

Ex.  29.  154  bu.  1  pk.  6  qt.  =  154.4375  bu. 

$j73.7£>a^     $m.74x^^     ^^3^^       ^^^^ 
154.4375  30.8875 

Ex.  30.  .0125  T.x  2000  =  25  lb.,  Ans. 

Ex.  31.  j\  of  2  bu.  3  pk.= Jl  pk. ;  3  bu.=12  pk. ; 

^1^12  =^^\,  ^715. 

Ex.  3^  4-  X I  X  I  X  -i-V-^  X  2  K=H?^^=1'747V,  Ans. 

Ex.  33.  V-  X  J/ x^3Q-xi-V-^x^yio,T  =  Vr32'=  301.339  f , 

Ans 

„     ^^    $.26x36x20     ^^^^^     . 

Ex.  34. =$20.80,  Ans. 

y 

^     ^      46x20x2x144 

Ex  35. =  13248,^715. 

4x5 

<ix.  86.  1864  yr.  4  mo.  20  da.  18  h.  30  min 
1836  "  9  "   4  "   3  "  45  " 


27  vr.  7  mo.  16  da.  14  h.  46  min. 


84  COMPOUND   NUMBERS. 

Ex.  37.   28  ft.  9'  =  28f  ft. ;  22  ft.  8'  =  22|  ft. ;  1  ft.  6=1^  ft 
i^i  X  -V  X  y  X  2V=-H-=1S1 5V  cu.  yd.,  Ans. 

Ex.  38.  30  bu.  54  lb.  =  30.9  bu. 

$1,375  X  30.9c^$42.4875,  Ans, 

Ex  39.  24  ft.  8'==:24|  ft. ;  18  ft.  6'=18i  ft. ; 

Ex.  40.  54  bu.  8  lb.=:54^  bu. 

$.84x541  =$45.50,  ^715. 

Ex.  41.  18720-r-120=:156,  Ans. 


^     ,^    21x5280x12      ^^^^^     . 
Ex.  42. =44352,  Ans. 

30  ' 


Ex.  43.  90^=324000'';  3'  12^  =  192'; 
324000-^192  =  1687|  mm.= 

1  da.  4  h.  7  min.  30  sec,  Ans. 


Ex.  44.  65  mi.=65  x  63360  in. ;  9  ft.  2  in.=110  m.; 
65x63360 


=  65x576  =  37440,  Ans. 


110 


Ex.  45.  10  bu.  X  2150.4  =  21504  cu.  in. 

21504x4     4096     ^^^4     ^ 
21504^571 =--3-^=.— =372-  qte. 

$.22  X  372yV=$81.92  sold  for  ; 
$  5    X    10     =$50.00  cost. 


$31.92  gain,  Ans, 


„     _    240x6x3x1728     ,,,,,^,    ., 

Ex.  46 ■ — =116640  bricks; 

8x4x2 


$3.25  X  116.64=$379.08,  Ans. 


PERCENTAGJfl. 


85 


Ex.  I 


PERCENTAGE. 

(S34, 

,  page  206.) 

.03 

Ex.  2. 

.0625 

.06 

.0875 

.09 

.3333  + 

.14 

.075 

.24 

.104 

.40 

.09625 

1.125 

1.035 

1.50 

2.25 

Ex.  3. 


Ex    4        -3-  •       J-  •      -3-  •       -O-  •       14.1  •       113.  • 

^^»   '*'         SOI         Js>         26»         200)         800)        tof) 


.0025 

.0075 

.00666  + 

.008 

.00625 

.0125 

.028 

.04333  + 

.0575 

.07125 

.122 

.25375 


11  3  .* 

4  0  0) 


^8  . 
26  ) 


(S355  page  207.) 

Ex.    9.   ^W5.  63  sheep.  Ex.  10.   Ans,  620  men. 

Ex.  12.   Ans,  $22692.25. 

Ex.  20     1.00  — .25=:.75;  760  sheep  x  .75=570  sheep,  ^w« 

Ex.21.    .18+.30=:.48;  1.00-.48=.52; 

$24500  X  .52=$12740,  Atis. 

1576  barrels x.l 2 5  =  197  barrels,  Ans. 


Ex,  22 
Ex.2? 


.75=1;   .33i=i; 


^  — 1= 


1^) 


$2760  X  y\=$1150,  A71S. 
Ex.24.     jXyVo=/5  sold;   |-^\=i|  left,  ^n«. 


86  PERCBNTAGB. 

Ex.  25.     I,  owed  after  the  1st  payment, 

fxf,         "         "       *'    2d 
I  X  f  X  I,  "         "       "    3d         " 
$^f  ^  X  f  X  f  X  ■i=$226.5G|,  Ans. 

(S36,  page  208.) 

Ex.  2.     90-^450  =  .20=:20  per  cent.,  Ans, 
Ex   3      175^1400=.125  =  12i  per  cent.,  Atv*. 
Ex.  4.     165-v-'750r=.22  =  22  per  cent.,  Ans. 
Ex.  5.     13.20-^240=. 055  =  51  per  cent,  Ans, 
Ex.  6.     .15-f-2  =  .075  =  7^  per  cent.,  Ans, 
Ex.  1.     6  bu.  1  pk.=200  qt;  4  bu.  2  pk.  6  qt.=  150  qt. 
150^200  =  .'75  =  75  per  cent.,  Ans, 
Ex.  8.     15  lb.  =  240  oz. ;  5  lb.  10  oz.=90  oz.  ; 

90-r-240  =  . 375  =  37^  per  cent.,  Ans, 
Ex.  9.     40-T-250=.16  =  16  per  cent.,  ^?i5. 
Ex.  10.     100  +  90  =  190; 

190-f-760=.25  =  25  per  cent.,  Ans, 

Ex.  11.     I  of  f =|=.50=50  per  cent.,  Ans 

(237,  page  209 ) 

Ex.  2.  16^.08  =  200,  Ans, 

Ex.  3.  42-r- .07=600,  Ans, 

Ex.  4.  75-^.125  =  600,  ^?i5. 

Ex.  5.  33^.0275  =  1200,  Ans. 

Ex.  6.  $281.25-^.375  =  $750,  Ans. 

Ek,  7.  50-^.20  =  250,  Ans, 

Ex.8.  $59.75-T-.125  =  $478,  ^W5. 

Ex.  9.  $975^.15=16500,  Ans. 

Ex.  IC  .40x.25=.l 

$1246.50-^.l  =$12465,  Ans, 

Ex.11.  2000-r- .40  =  5000;  5000-2000  =  3000,  Jn.<f. 


Ex. 

2. 

Ex. 

3. 

Ex. 

4. 

Ex. 

5. 

Ex. 

6. 

Ex. 

7. 

co3iMissioi;r  akd  brokerage.  87 

(238,  page  211.) 
1.00-f  .18  =  1.18;   14754-1.18  =  1250,  Ans, 
1.00-h.25=3l.25  ;  $4.00-^1.25  =  $3.20,  Ans. 
1.00  +  . 15  =  1. 15;  $6900-T-1.15  =  $6000,  Ans. 
1.00  +  .08,V-1.08A^; 

$432250^1.080625  =  1400000,  Ans. 
1.00  +  .041  =  1.0425  ; 

$8757^1.0425=18400,  Ans. 
Since  he  increased  his  capital  the  first  year  by 
20  %  of  itself  he  mast  have  had  100  ^  +  20  ^,  or 
120  %  of  original  capital  for  new  capital  the  sec- 
ond year ;  and  since  he  increased  his  new  capital 
by  20  %  he  must  have  had  120  %  of  120  %,  that 
is,  144  ^,  of  original  capital;  therefore 
$9360-f-1.44  =  $6500,  Ans. 

(239^  page  212.) 

Ex.2.     1.00-.15  =  .85;  340 -f-. 85  =  400,  Ans. 

Ex.3.      1.00  — .20  =  . 80;   $1000-^.80  =  $1250,  ^7i5. 

Ex.4.     1.00-.24  =  .76;  $4028 -f-. 76  =  $5300,  ^^5. 

Ex.  5.     1.00-.00l=.995  ;  298^--.995  =  300,  Ans. 

Ex.  6.  $198,  his  selling  price,  is  90  ^  of  his  asking  price ; 
therefore  $198-^.90  =  $220,  his  asking  price;  and 
$220,  his  asking  price,  is  110  ^  of  the  cost ;  there- 
fore $220-f-1.10=$200  cost,  Ans. 


COMMISSION   AND   BROKERAGE. 


(243,  page  213.) 
Ex.  2.     $6756  X  .02=$135.12,  Ans. 
Ex.  3.     $17380  X  .035  =  $608.30,  Ans. 


87a  PEECEKTAGE. 

Ex.  4.     $.75x4700=$3525;$3525x.015=$52.875,^n5. 
Ex.  5.     $25875  x  .0075  =  164.6875,  Ans, 
Ex.6.     $32844-12176. 50  =  $5460.50; 

$5460.50  x.0225=:$122. 86 +,  ^W5. 

Ex.  7.     $2890  X  54^=:$23.12,  Ans. 
Ex.  8.     $.32  X  26750=$8560  ; 

$8560  X  .02|  =  $235.40,  Ans, 
Ex.  9.     400  X  570  x  $.09  x  .0225  =  $461.70,  Ans. 
Ex.  10.  $7.60x450  =$3420 

.25x56x38    =      532 

.09x48x105=     453.60 


$4405.60  X  .055  =  $242,308,^/15. 
Ex.11.  $950x.06i  =  $61.75,  fee;   $950— $61.75 

=  $888.25,  remitted. 
Ex.12.  $30456.50  x.06  =  $1827.39 
19814.15 


$10642.35  X  .04=     425.694 


«  $2253.084,  Ans. 

(244,  page  215.) 
Ex.  2.     $3246.20^]. 02  =  $3182.55  (nearly)  invested; 

$3246.20  — $3182.55  =  $63.65,  Ans. 
Ex.  3.     $9682-M.03  =  $9400,  Ans. 
Ex.  4.     $10246,50-^1.035  =  $9900  invested  ; 

$9900-^$5.50  =  1800,  Ans. 
Ex.  5.     $4908^1.045  =  $4695.69  +  ,  Ans. 
Ex.  6.     $603.75-7- 1.05  =  $575  invested; 

$575-^$5  =  115,  Ans. 

Ex.  7.     .03 +  .015  =  . 045 

$9376.158-+1.045  =  $8972.40,  to  pay  out; 
$9376.158  — $8972.40  =  $403.758,  fees,  Ans. 


STOCKS.  876 

Ex.  8.     $13842.0'7-^1.01Y5=:$13604,  invested  ; 

$13842.07  — $13604=$238.07,  commission,  Ans. 
Ex.  9.     $10650^1.0025==$10623.44  +  ,  Ans, 


STOCKS. 

(261,  page  218.) 
Ex.2.     $1200x.95  =  $114'0,  ^715. 
Ex.  3.     $3500  X  .85=^$2975,  Ans. 

Ex.  4.     $1.00  +  $.05l+$.00l=:$1.06,costofdoll.ofstock; 

$150x48  =  $7200,  nominal  amount; 

$1.06  X  7200=:$7632,  ^?Z5. 
Ex.  5.     $1.09|  X  5364=:$5853.465,  Ans. 
Ex.  6.     $6275  X  .12=:$753,  Ans. 
Ex.  7.     1.00 +  .04|  +  .00|r=  1.05; 

$25000  X  1.05=r$131250,  Ans. 
Ex.  8.     .14 +  .125  =  .265,  rate  of  gain  ; 

$4200  X  .265  =  $11 13,  Ans. 
Ex.  9.     $17500  X  1.0075  =  $l7631.25,  Ans. 
Ex.  10.  .03 +.0225  =  .0525,  rate  of  gain; 

$50  X  75 =$3750,  nominal  amount  of  stock  ; 

$3750x.0525=$196.875,  Ans. 

Ex.11.  $50x28  =  $1400;  $1400  X  1.07  =  $1498  ; 

$100x25  =  $2500;   $2500X.875  =  $2187.50; 
$2187.50  — $1498=:$689.50,  Ans. 

Ex.  12.  $3600x.95  =  $3420; 

$2700xl.03=:$2781;$3420-$2781=$639,^n^ 

Ex.  13.  $12000  X.145  =  $l  740,  Ans. 

(262,  page  219.) 
Ex.  2.     $6300  =  1. 05  =  $6000  =  60  shares,  Ans. 
Ex.  3.     $6187.50-+.90=$6875,  Ans. 


88  PERCEISTAGE. 

Ex.  4.     $53500-^1. 07  =  $50000,  Ans, 
Ex.  5.     $1150-^-.92  — $1250,  nominal  amount; 
$1250-^50:=$25,  Ans, 

(265,   page  222.) 
Ex.  2.     #867^1.02=z$850,  stock  purchased  ; 

$850  X  .06  =  $51,  income,  Aiis. 
Ex.  3.     $8428-^.98  =  $860Q,  stock  pm'chased; 

$8600  X. 05  =  1430,  income,  Ans. 
Ex.  4.     1.04|-f  .00|  =  1.05; 

$10500-^1.05c=:$10000,  Ans, 
Ex.5.     .87  +  .00i  =  .875  ; 

$4795-f-.875  =  $5480,  stock  purchased ; 

$5480  X. 05  =  $274,  income,  Ans, 

Ex.  6.     1.07i  +  .00i-  =  1.08  ;  .96i  +  .00-i=.97  ; 
$10476-M.08=:$9700,  purchase  of  G's  ; 
$10476-^.97  =  $10800,         "         "  5-20's; 
$9700  X  .06   =$582,  income  from  6's ; 
$10800  X. 05  =  $540,       "         "      5-20's; 

$42,  Ans. 
Ex.  7.     1.08|  +  .00|  =  1.09; 
$125xl09  =  $13625; 
$13625-M.09  =  $12500,  stock  purchased; 
$12500  X  .06 =$750,  income  from  stock  ; 
$750  — $681.25=$68.75,  income  increased,  Ans 

(266,  page  223.) 
Ex.  2.     $840-^.06 =$14000,  stock  required  ; 

$14000  X  .95  =  $13300,  investment,  Ans. 
Ex.  3.     $1860-^.05  =$37200,  stock  required. 

$37200  X  .98^  =  $36642,  investment,  Ans. 
Ex.  4.     $1 080 -f-. 05  =  $21600,  stock  required; 

$21600  X  1.08l  =  $23436,  investment,  Ans. 


GOLD   IKYESTMEKTS.  88a 

Ex,  5.     $25000  x.93|  3=123437.50,  proceeds  of  5-20's  ; 
$960^.06  =  116000,  6's  required  ; 
$16000  xl.OQi  =$17480,  investment  in  U.  S.  6's  I 
$23437.50-$l7480  =  $5957.50,costofhouse,^w5. 

(267^  page  .223.) 
Ex.  2.     .06-^.87=:.06|g=:6ff  %,  -^ns. 
Ex.  3.     .06-M.05  =  .05f  =:5|  %,  Ans. 
Ex.  4.     .06^.75  =  .08=:8  ^,  Ans. 
Ex.5.     .06-^1.08=:.05|  =  5|  ^,  ^715. 
Ex.  6.     .05-.985  =  .05J/^  =  5J^,  %; 

,0Q-~1.0d  =  .05j%%  =  5j%%  %,  better,  Ans. 

(268^  page  224.) 
Ex  2.  .06-^.09  =  .66|  =  66|  ^, -^715. 
Ex,  3.     .06-f-. 05  =  1.20; 

1.20  — 1.00  =  .20  =  20  fo  premium,  Ans. 
Ex.  4.     .05^.06  =  .83l  =  83l  %,  ^ns. 
Ex.  5.     .05^.07  =  . 7l|; 

1.00  — .711  =  . 28|  =  28|  ^,  Ans. 


GOLD   Il^rYESTMEKTS. 


(270^  page  225.) 
Ex.  2.     $1.47  X  150  =  $367.50,  Ans. 
Ex.  3.     $1.37|  X  320.50  =  $440.68|,  Ans. 
Ex.  4.     $1.33  X  2500  =  $3325,  Ans. 
Ex.  5.     $8000  X  .05  =  $400,  income  in  gold  ; 

$1.38x400  =  $552,  "      "  ciivvency,  Ans. 

Ex.  6.     $9500  X  .06  =  $570,  income  in  gold  ; 

$1.40x570  =  $798,         "      "  currency,  ^^^5. 


886  PEECEKTAGB. 

Ex.  1.     $1,475  X  3000=$4425,  cost  of  house  in  currency 
^       by  latter  offer  ; 
$4500  — $4425  =  $75,  gain,  Ans. 

271,  page  226.) 

Ex.  2.     $1.00^$1.38i=r:$7225^6_^  Ans. 
Ex.  3.     $].00-~$1.45i=.68||  ; 

1.00  — .68||=.31^iy=:31^^y  %,  Ans.  to  first. 

$1.00-^$1.47  =  .68y|^; 

1.00-.683|^=r.31j||=:311||^,^w5,  to  second. 

$1.00-^$1.955  =  .5l3W  ; 

1.00-.51-3\\=3.48||f  =  48|§f  %,  Ans,  to  third. 

$1.00^$2.85-— .35|; 

1.00  — .35|  =  .64f  =  64|  %t  ^^s.  to  fourth. 

Ex.  4.     $4181-^$1.48=:$2825,  Ans. 
Ex.  5.     $.24-^$1.60  =  $.15,  Ans. 

Ex.  6.     $5900  X  .90  =  $53 1 0,  proceeds  of  sale  of  10-40's  ; 
$5310-^$1.47^  =  $3600,  gold  purchased,  Ans. 

Ex.  7.     $1.00^$1.45=$.68|f,  Ans. 

Ex.  8.     $792-f-$1.65  =  $480,  Ans. 

Ex.9.     $1.00-^$.30:=-3.33^    ==333^     ^,  /4w5.  to  first. 

$1.00-^$.45  =  2.22|    =222f     ^,  *•  "  second. 

$1.00^$.54=:1.85f^  =  1853/^^,  "  "  thu'd. 

$1.00-r-$.60  =  1.66|    =:166f     %,  "  "  fourth. 

$1.00~$.74=rl.353\  =  135/^  ^,  *•  "fifth. 

Ex.  10.  $126-^$1.40=$90,  value  of  currency  in  gold; 
$90^$3.50  =  25f  yards,  Ans. 

Ex.11,  $11.75-^$1.50=|7.83l,pu^ch.  price  of  flour,gold; 
$10.35-^$1.35=:S7.66|,  selling     "      "      "       " 
$7.831 —  $7.66|  =  $.161,  loss  in  gold  on  one  barrel ; 
$.16§  x300  =  $50,  entire  loss,  Ans. 


PROFIT  Ais^D   LOSS.  .    89 

Ex  12.  $.07-^$1.40  =  .05  =  5   %,  rate  of  income  in  gold 
from  mortgage ; 
.06  — .05=:1  ^,  5-20's  better,  Ans. 

Ex.  13.  $51100  X. 073=:  $3730.80,  inc.  from  Y.SO's,  in  cur.; 
$3'730.30^$1.46=$2555,  '"       "     '/.30's  in  gold; 
$51100  X  1.04  =$53144,  proceeds  of  7.30's; 
$53144-^$1.46  =  $36400,  gold  purchased; 
$36400-f-$.70  =  $52000,  10-40's  purchased; 
$52000  X  .05  =  $2600,  inc.  from  10-40's  in  gold; 
$2600  — $2555  =  $45,  income  increased,  Ans. 


PROFIT  AKD  LOSS. 
(27  3^  page  228.) 

Ex.  2.     $84.80  X  .125  =  $10.60,  Ans. 

Ex.  3.     $1.15  x500=:$575,  cost  of  wheat; 
$575x.l6~$95.83i,  Ans, 

Ex.4.     $3,625  X  76  =$275.50,  cost  of  wood;  * 

$275.50  X  .26=$71.63,  Ans, 

Ex.  5.     $1.75  X  40  =  $70,  cost ;  $70  x  .14f =$10,  Ans, 

Ex.  6      $.0825x230  x3  =  $56. 925,  cost;  .18y\  =  -,\; 
$56,925  X  j\  =$10.35,  gain  ; 
$56,925  +$10.35  =$67,275;  230  lb.  X  3  =  690  lbs. 
$67.275 -f-690  =  $.0975,  selling  price,  Ans. 

Ex.  7.     $.625  X  3840  x  .375  =  $900,  Ans. 

Or  $.625=$^;  .375  =|;  ssjj).  x  |  x  |  =  $900,  ^/25. 


90  PERCENTAGE. 

Ex.  8.     $4720  X  .125=:$590,  loss  in  the  bargain ; 
14720— $590=$4130 ;  $4130  x  .15=$619.50  loss  in  bad  debts 
$590-l-$619.50=$1209.50,  Ans. 

Ex.  9.  1  +  .225  =  1.225,  bis  per  cent,  after  1  year  ; 

1.225  X  1.30  =  1.5925,  bis  per  cent,  after  2  years  ; 
1.5925  X  I  =1.3270f,  bis  per  cent,  after  3  yeara 
$3000  x.3270f =$981.25,  Ans. 

(27  4,  page  229.) 

Ex.  2.     $330— 1275  =  $55,  gain  ; 
$55-^$275=.20,  Ans. 

Ex.  3.     $.75— $.60=$.15,  gain  ; 
$.15-^$.60=.25,  Ans. 

^     ^  $114,885  ^^      , 

Ex.4.     -    ^     ^,  ^^   —.23,  Ans. 
108  x  $4,625  ' 

Ex.  5.     $.095— $.08 =$.01 5,  gain  on  1  lb.  , 

$.015-^$.08=. 1875  =  183  per  cent.,  Ans. 

Ex.  6.     $42  X  150=$6300  ;  $6300— $5400=$900- 
$900-r-$6300=.14f,  Ans. 

Jfix.  7.     $25— $15=$10  ;  $10-t-$25=.40,  Ans. 

Ex.  8.     $.25  X  20=$5.00,  received  per  ream  ; 
$5.00— $2.00  =  $3.00,  gain  per  ream  ; 
$3.00-^-$2.00  =  1.50=150  per  cent.,  Ans. 

Ex.  9.     If  I  sells  for  J  its  cost,  1   sells  for  |  =  |  its  cost 
f  — 1=1^,  gain  on  1,  =.50  =  50  per  cent.,  Ans. 

Ex.  10.     i-T-f =f  ;  or  tbe  whole  would  be  sold  for  f  of  itfl 
cost ;  hence  f  of  the  cost  Avas  lost.      And, 
1  =  371  per  cent.,  Ans. 

Ex.  11,     One  peck  is  gained  on  3  pecks  ;  hence 
l-f-3=.33i  per  cent.,  Ans, 


PKOFIT    AND    LOSS.  91 


Ex.  12. 

8  7  J-  per  cent.,  Ans. 

Ex.  13. 

342  lb.  @$.08  ==$27.36 

3781b.  @    .081=  32.13 

$59.49  sold  for ; 

720  lb.  X  .07     =  60.40  cost ; 

$9.09  gain  ; 
$9.09-^$50.40=.182V  per  cent.,  Ans, 
Ex.  14.     $1.60  — $1.25=1.35,  gain  per  gal. ; 
$.354-$1.25=.28,  gain  per  cent. ; 
$1.25  X  63  X  2  X  .28 =$44.10,  whole  gain,  Ans, 

Ex.  15.     $.66— $.55  =  $.ll,  gain  per  bushel  on  the  corn  ; 
$.ll-f-$.55=.20,  gain  per  cent,  on  the  corn. 
$1.375 -$1.10=$.275,  gain  per  bushel  on  the  wheat  , 
$.275-r-$1.10  =  .25,  gain  per  cent,  on  the  wheat; 
.25— .20=. 05  per  cent,  on  the  wheat,  Ans. 

(275,  page  231.) 

^     ^      $140x1.25     ^^^     , 

Ex.  2. =$.14,  Ans, 

1250  ' 

Ex.  3.     $.30  X  1.16|  X  1.33i=$.46|,  Ans. 

Ex.  4.     $.105  X  l.l7i=$.1232,  Ans. 

Ex.  5.     1.00-.15  =  .85 

$.62ix.85=$  .531  y 
1.20    x.85=   1.02     I  Ans. 
3.875  X. 85=   3.29f  ) 
Ex.  6.     $3240  X  .82  =  $2656.80,  Ans. 

Ex.7.     $28xl20=$3360;    $3360  + $480  =  $3840,  whole 
cost;  $3840xl.l2i=$4320;  $4320  — $3840=$480,  gain  ; 
$4320-^120  =  $36,  an  acre,  Ans, 
Ex.  8.     $2.60  X  52  =  $135.20  ;  $135.20  x  If =$185,90  ; 
52-^7  =  45;  $185.90-4-45=$4.13i,  ^?i5. 


92  PERCENTAQB. 

Ex.9.     1.18f  =  if;    23i=-V-; 

$3.±s.  X  If  X  j\  X  yV=82.85,  An9. 

(276,  page  232.) 

Ex.  2.  $.08-^.80  =  1.10,  Ans, 

Ex.  3.  $6.125-^.875  =  $7.00,  ^/i5. 

Ex.  4.  $.96-^1.28  =  $.75,  Ans. 

Ex.  5.  1.18Jr=if  ;  $1881-^if  =  $1584,  Ana. 

Ex.6.       $69.75       ^^^,     . 

Ex.  7.     $ V- X -y- X  VV=$86.25,  ^W5. 

Ex.  8.     $96-^80=$120,  cost; 

$120xl.l5  =  $138,  Ans. 

Ex.9.     1.125  =  1;  1.18f  =  if  ; 

$570  X  If  X  f =$426,661,  Ans. 

Ex.  10.  $24x4=$96  whole,  proceeds. 

$24  X  2=$48  ;  $48-M.20  =  $40,  cost  of  1st  2  bW. 
$48-^  .80  =  $60,  cost  of  2d  2  bbi. 
$40  +  $60  =  $100;  $100— $96=$4,  lost. 

Ex.  11.  $4900-f-1.40=$3500  =  3timeswhatLe  began  with; 
and  $3500-r-3=$1166.66|,  Ans. 


mSURANCK 


(282,  page  234.) 

Ex.  2.     $750  X  .04=$30,  Ans. 

Ex.  3.     $4572.80  x  .025=$114.32,  Ans. 

Ex.  4.     $5700  X  .01 75  =  $99.75,  Ans. 


TAXES.  98 


Ex.  5.     $28400  X  .035  =  $994,  Ans, 

Ex.  6.     $55800  X  .028zr:$1562.40,  premium  ; 

$55800 -$1562.40z=$54237.60,  Ans. 

Ex.  7.     $47500  X  ^-f  ^==$356.25,  Ans, 

Ex.8.     $8000  + 14000=:$12000; 

$12000  X  .021  =  8285,  Ans, 

Ex.  9.     $1.20  X  4000=$4800,  worth  of  wheat; 
$4800  X  I =$3200,  amount  insured  for; 
$3200  X  f  X  2? 0  —  *3^>  premium  ; 
$3200— $36==$3164,  saved  by  insuring; 
$4800  — $3164=$1636,  owner's  loss,  Ans, 

Ex.  10.  $21000  X  5^0=^^168; 

$15400x^^0=^  9^-25; 

$264.25,  Ans, 


TAXES. 
(289,   page  23G.) 


Ex.  3.     Property  tax  ==$26.95 
1  poll  .75 

$27.70,^^15. 

Ex.  4.     $.50  X  2981  =$1490.50,  poll  tax  ; 

$9190.50— $1490. 50=$7700,  properly  tax  ; 
$7700-^-$l,400,000  =  .0055,  rate  of  taxation. 
$12450  x.00553=$68.475 
2  polls  =$  1.00 

C's  tax  $69,475,  Ans, 

Ex.  5      $5375  X  .0055  =$29.5625,  Ans. 
K.  p.  5 


M  PERCENTAGE. 

Ex.  6.     $.625  X  30  =  ^18.75,  poll  tax  ; 

$4342.75— $18.75  =  $4324,  property  tax; 
$4324-^$188000=:.023,  rate  of  taxation. 
$2500  X. 023 +  $.625  =$58,125,  Ans. 

Ex.  7.     $.30  X  25482  =  $7644.60,  poll  Jpx  ; 

$103294.60  — $7644.60=$95650,  property  tax  ; 
$95650-J-$38260000=.0025,  rate  of  taxation. 
$9470  X. 0025 -f-$.90  =  $24.575,  Ans. 

Ex.  8.     $10000  X  1.025  =  $10250,  whole  tax; 
$1.25x225=:$281.25,  poll  tax ; 
$10250-$281.25=$9968.75,  property  tax; 
$9968.75-^$1246093.75=:.008,  rate  of  taxatioL  . 
$11500  X  .008+ $1.25  =  $93.25,  E's  tax,  Am. 

Ex.  9      $275.57  — $98=$177.57,  tax  ; 

$l77.57-^3946=$.045,  rate  per  day; 
$.045  X  118  X  2=$10.62,  Ans. 


CUSTOM   HOUSE   BUSINESS.  95 


CUSTOM  HOUSE  BUSINESS. 


(303,  page  239.) 


Ex.  2.     $-95  X  224=$212.80,  value  of  the  silk , 
$212.80  X.19=$40.432,  Ans. 


Ex.  3.     $.54  X  31.6  X  50=$850.50,  gross  value  ; 
deduct  $850.50  x  .02=5     17.01  for  leakage ; 
$833.49,  net  value; 
$833.49  X  .24 =$200.0376,  Ans. 


Ex.  4.    $.15  X  115  X  175=$3018.75,  value  of  the  coflfU? 
$3018.75  X  .15=$452.81J,  Ans. 


Ex.  5.     $.36  X  63  X  25 =$567,         gross  value ; 
deduct  $567  x  .005=       2.835,  for  leakage  ; 
$564,165,  net  value; 
-^        $564,165  X  .24=$135.3906,  Ans. 


PERCENTAGB. 


SIMPLE    INTEREST. 

(311,  page  241.) 

Ex.  3.     $45.92,  Ans,  Ex.  8.     $093.83+,  Ara 

Ex.  11.  $607.50,  Ans,  Ex.  15.  $440,625,  Ans. 

Ex.  17.  $605.70  +  liit.  for  3  yr.=$751.068,  Ans. 


(313,  page  245.) 

Ex.  7.     $106,855,  Ans.  Ex.  8.     $1.72+,  Ans. 

Ex.  11.  $91.85  +  ,  Ans.  Ex.  15.  $2,138+,  Ans.  ' 

Ex.  18.  $24.87+,  Ans.  Ex.  19.  $282.75+,  Ans. 

Ex.  22.  $82.36+,  Ans. 

Ex.  23.  Time,  7  yr.  7  mo.  2  da.;  $51.98  +  ,  Ans. 

Ex.  24.  Time,  2  yr.  1  mo.  4  da. ;  $4,474,  Ans. 

Ex.  25.  Time,  11  yr.3mo.27  da.;  $19.818f +  ,  Ans. 

Ex.  26.  Time,  9  mo.  19  da.;  $408,957  +  ,  Ans. 

Ex.  27.  First  payment,       $2000 

Second  payment,  $3157.50 
Third  payment,     $1105. 


$6262.50,  Ans. 


Ex.  28.  $350.  +  int.  for  11  mo.  21  da.=  $373,887  + 
$150.  +int.  for  8  mo.  16  da.=  157.466  + 
$550.50  +  int.  for  3  mo.  11  da.=     561.310  + 

Total  =  $1092.663  + ,  Ansi 


PARTIAL   PAYMENTS. 


97 


PARTIAL  PAYMENTS. 
(314,  page  249.) 

Kx  3.  Amt.ofnotetoNov.l2,  1858,  (4mo.  22da.)$535,27-f" 
Payment, 105.50 

New  Principal, $429.77-+- 

Amt.,  Mar.  20,  1860,  (16  mo.  8  da.) 488.03  + 

Payment, 200 

New  Principal, $288.03  + 

Amt.,  July  10,  1860,  (3  mo.  20  da.) 296. 83 -j 

Payment, 75.60 

New  Principal, ..$221.23  + 

Amt.,  June  20,  1861,  (11  mo.  10  da.) $242.12+, 

Ann 

Ex.  4.  Amt.  of  note,  Jan.  1,  1860,  (7  mo.  24  da.)  $3136.50 
Sum  of  payments  to  tliis  date, 525.00 

New  Principal, $2611.50 

Amt.,  April  4, 1861,  (15  mo.  3  da.) 2841.53  + 

Sum  of  payments, 1575.00 

' New  Principal, $1266.53  + 

Amt.,  Feb.  20,  1862,  (10  mo.  16  da.) $1344.35+, 

Ex.  6.  Amt.  of  note,  Jar..  1,  1852,  (16  mo.  28  da.)$977.15  + 
Payment, 250.00 

New  Principal, $727.15  - 

Amt.  May  4,  1853,  (16  mo.  3  da.) 775.93  + 

Payment, 316.75 

New  Principal $459,18  4 

Amt.  Sept.  15,  1853,  (4  mo.  11  da.) $467.53+, 

5  Ans 


98  PERCENTAGE. 

Ex.  6.  Interest  commence  1  Aug.  2,  1860. 

Amt.  of  note,  May  6,  1861,  (9  mo.  4  da.),. $192.988 -f 
Payment, 50 

New  Principal, $142,988  + 

Amt.  Aug.  26,  1862,  (15  mo.  20  da.) $154,188+. 

Ex.  1.  Amt.  of  mortgage,  Jan.  1,  1852,  (3  mo.).  .$6120.00 
Payment, 500 

New  Principal, $5620.00 

Amt.  Sept.  10,  1852,  (8  mo.  9  da.) 5930.98  + 

Payment, 1126.00 

New  Principal, $4804.98  + 

Amt.  March  31,  1854,  (18  mo.  21  da.)  . . .   5404.00 -f- 
Payment, 2000.00 

New  Principal, $3404.00  + 

Amt  Aug.  10,  1854,  (4  mo.  9  da.) 3501.57  + 

Payment, 876.50 

New  Principal, .$2625.07  + 

Amt.  Oct.  1,  1857,  (37  mo.  21  da.) $3284.84  + 

(315,  page  251.) 

Ex.  1,  Amt.  from  Jan.  1,  1858,  to  Jan.  1,  1859, 

(1  yr.) $487.60 

Amt.  of  1st  pay't  from  Apr.  16,  1858,  to 

Jan.  1,  1859,  (8  mo.  15  da.) 154.29 

New  Principal, $333.31 

Amt.  from  Jan.  1,  1859,  to  Mar.  11,  1860, 

(14  mo.  10  da.) 357.19  + 

Payment, 75.00 

New  Principal, $282.19  + 


PARTIAL    PAYMENTS.  H 

Ami.  from  Mar.  1 1,  1860,  to  Dec.  11, 1860, 

(9  mo.) $294. 89  + 

Amt.  of  3d  pay't  from  Sept.  21,  1860,  to 

Dec.  11,1860,  (2  mo.  20  da.) 56.74  + 

Ans,  $238.15 -h 

(316,  page  251.) 

Ex.  1.  Amt.  of  Principal,  Jan.  1,  1859, 

(2  yr.  8  mo.  20  da.) $698.00 

A.mt.  of  1st  endorsement,  (for  2  yr. 

4  mo.  21  da.) $178,386 

Amt.  of  2d  endorsement,  (for  1  yr. 

10  mo.  19  da.) 222.633 

Amt,of3dendorsement,(for7mo.)   191.475     592.494 


Ans.  $105.50  +  , 


(SIT,  page  252.) 


Ex  1.  Amt.  of  note,  Aug.  4,  1859,  (1  yr.) $609.50 

Amt.  of  1st  pay't,  Aug.  4,  1859,  (9  mo.) . .      66.88 


New  Principal, $542.62 

Amt.  of  new  Principal,  Aug.  4, 1860,  (1  yr.)    575.17 
Amt.  of  2d  pay't,  Aug.  4,  18G0, 

(7  mo.  21  da.) $49.85 

Amt.  of  3d  pay't,  Aug.  4,  1860, 

(4  mo.  18  da.) 253.70        303.55 


New  Principal, $271.62 

Amt.  of  new  Prin.,  Nov.  4,  1860,  (3  mo.).    275.69 
Amt.  of  4tli  pay't,   Nov.  4,    1860,    (1  mo. 

6  da.) 60.30 

Ans,  $215.33. 


100  PERCENTAGE. 

(31 85   page   253) 

Ex.  1.  1st  installment  of  interest,  due  Feb.  2,  1856,  ^30 

2d           "           "         "         ''       "     "    1857,  30 

3d           "           "         "         "       "     «    1858,  30 

4th          «           "         "         "       "     "    1859,  30 

5th         "           "  .       "         "     Aug.  2,  1859  15 

4135 

1st  installment  draws  int.  3  jr.  6  mo. 
2d  "  "         "     2  jr.  6  mo. 

3d  "  "         "     1  jY,  6  mo. 

4th  "  "         "  6  mo. 


Int.  of  $30  for  8  yr.  0  mo $14.40 

Principal, $500.00 

Ans,  $649.40 


PROBLEMS    IN    INTEREST. 


(320.,  page  253.) 

Ex.  2.     Int.  of  $1  for  6  yr.  3  mo.  at  6  per  cent.,  is  $.395 ; 

$28.125^.375  =  $75,  Ans. 

Ex.  3.     Int.  of  $1  for  4  mo.  18  da.  at  4  per  cent.,  $.015^  ; 

$9.20-^.015i=z$600,  Ans. 
Ex.  4.  $1260^.07  =  $18000,  Ans. 
Ex.  5.     $33'70-^.10=z:$33'700,  Ans. 

(321,  page  254.) 

Ex.  2.     $1  for  8  mo.  at  6  per  cent.,  amounts  to   $1.04} 

$655.20-^1.04=:$630,  Ans. 

Ex.  3.     Amt.  of  $1   for  5  yr.  5  mo.  9  da.  at  5  per  cent., 
1.27201  ;  $106.855 4-1.2720f  =$84,  Ans. 


.^tyh 


PROBLEMS   IN    INTEREST.  101 

Ex.  4.     Amt.  of  $1  for  8  yr.  5  mo.  at  5  J  per  cent., 
$1.462916+  ; 

$1897.545^1.462916  + =$1297.09+,  Ans. 

Ex.  6      Amt.  of  $1  for  3  yr.  4  mo.  at  7  per  cent.,  $1.23^  ; 
$221.075-+1.23i  =  $l79.25,  Ans. 

Ex.  6.     Amt.  of  $1  for  11  yr.  8  da.  at  IQi  per  cent.,  $2.1 57 -J ; 
$857.54  +  2.157iirr$397.50,  principal ; 
Int.  of  $397.50  for  11  yr.  8  da.,  at  10^  per  cent.,  = 
$460.04,  Ans. 

(322,  page  255.) 

Ex.  2.     Int.  of  $500  for  3  yr.  at  1  per  cent.,  $15  ; 

$45 -+$15  =  3   per  cent.,  Ans. 

Ex.  3.     Int.  of  $180  for  1  yr.  2  mo.  6  da.  at  1  per  cent., 
$2.13  ;  $12.78-+$2.13=:6  per  cent.,  Ans. 

Ex.  4.     Int.  of  $2000  for  6  mo.  at  1  per  cent.,  $10  ; 
$75-^$10  =  7^  per  cent,  per  annum,  Ans. 

Ex.  5.     Int.  of  $1000  for  3  yr.  3  mo.  29  da.  at  1  per  cent, 
$33,305+  ; 
$183.18-+$33,305=:5.5  per  cent.,  Ans. 

Ex.  6.     Int.  of  $21640  for  1  year  at  1  per  cent,  $216.40  ; 
$2596.80-^$216.40=rl2  per  cent,  Ans. 

(323,  page  256.) 

Ex.  2.     $325  X  .06  =$19.50,  int  for  1  yr. ; 
$58.50-+$19.50  =  3  yr.,  Ans. 

Ex.3.     $1600x.06  =  $96;  $2000-$1600=$400; 
$400+-$96  =  4J  yr-=4  yr.  2  mo.,  Ans. 

Ex.4.     $204x.07--$14.28;  $217.09  — $204=$13.09; 
$13.09  -;  $14.28  =  f  J-  yr.  =  ll  mo..  Am, 


102  PERCENTAGE. 

Ex.  5.     $750  X  .Oe  =  $45  ;  $942  — S750=$192  ; 

$192-v-$45  =  4y4j  yr.  =  4  yr.  3  mo.  6  da.,  Am. 

Ex.6.     $200  X. 06 =$12; 

$200^$12=:16  I  yr.=16  yr.  8  mo.,  An8. 

Ex.7.     $675x.05=r$33.75; 

$675-^$33.75  =  20  years,  Ans. 


COMPOUND  INTEREST. 
(324,  page  257.) 


Ex.  2.     $500.00  Prin.  for  1st  year. 
35.00  Int.       "      "       " 


$535.00  Prin.    "     2d      ** 
37.45  Int.      "      "       ** 


$572.45  Amt.    "     2  years. 
600.       Given  Prin. 


Ajis,  $  72.45  Compound  interest. 

fix.  3.     $312.00      Prin.  for  1st  year. 
18.72      Int.      "     "      " 


$330.72      Prin.    "    2d      " 
19.84      Int.      "     "       " 


$350.56      Prin.    "    3d      " 
21.03      Int. 


Ans.  $371.59  +  ,  Arat.   "    3  year^ 


COMPOUND    INTEREST.  103 


Ex.  4.     $250.00     Prin.  for  1st.  half  year. 
7.50     Int.      "     "  " 


$257.50 

7.72 

$265.22 
7.96 

Prin. 
Int. 

Prin. 
Int. 

Prin. 
Int. 

Amt 

u 

u 

u 
u 

2d 
3d 

$273.18 
8.19 

4tli 
u 

$281.37 
250.00 

2  years. 

Ans,    $31.37  +  Compound  interest 

Ex.  5.     $450.00     Prin.  for  1st  quarter. 
7.87     Int.      "     "        " 


$457.87 

Prin.    " 

2d 

8.01 

Int.      " 
Prin.    " 

u 

$465.88 

3d 

8.15 

Int.      " 
Prin.    " 

(( 

$474.03 

4th 

8.30 

Int.      " 

(( 

^?6S.  $482.33+ Amt.    "    1  year. 

Ex.  6.     $236.00     Prin.  for  1st  year. 
14.16     Int.      "     "      " 


$250.16     Prin.   "    2d 
15.01     Int.      "     " 


$265.17     Prin.   "   3d 
15.91     Int.      "     " 


$281.08     Prin.    "  4th    •* 
16.86     Int.      «     «      - 


IGil  PERCENTAGE. 

1297.94     Prin.    "    1  mo.  G  d*. 
10.72     Int.      "    7     "    6    " 


$308.66     Arat.    "    4  yr.  7  mo.  6  dfi, 
230.00     Given  principal. 


Ans,  $72.66+ Int.  4  yr.  7  mo.  6  da. 

Ex.  7      $700.00     Prin.  for  1st  year. 
49.00     Int.      "     "      " 


$749.00  Prin.  "  2d  " 

52.43  Int.  "  "  " 

$801.43  Prin.  "  3d  " 

56.10  Int.  "  "  *' 


$857.53     Prin.    "  9  mo.  24  da. 
49.02     Int.      "    9    "     24    " 


Ans,  $906.55 -f,Amt."    4  yr.  9  mo.  24  da. 
Ex.  9.     $120  X  2.078928==$129.47  +  ,  ^/2S. 
Ex.10.  $.10x3.86968=$.386968,  ^^5. 


DISCOUNT. 


(326,  page  259.) 

Ex.  2.     $180-M.20r=:$150,  Ans, 

Ex.  3.     $1315.389^1.175  =  $1119.48,  Ans, 

Ex.  4.     $866.0384-1.281i  =  $675.888-f,pref. worth.  [ 

$866.038— $675.888 +  =$190.15+.  viuv-ft     ]      '** 

^x.  5.     $1005  — $475=$530 


$475-^1.05       =$452.38  + 
$530-^1.075   =   493.02  + 


$945.40  f ,  Ans. 


PROMISCUOUS   EXAMPLES,  10t5 

Ex.  6.     Term  of  discount,  6  mo.  24  da. 

$529.925-^1.034  =  $512.50  present  worth. 
$529.925  — $512.50  =  $17.425  discount,  Ans. 

Ex.  7.  $3675  cash  offer. 

$4235-^1.21=$3500  cash  value  of  note. 


Am,  $  175,  loss. 

Ex.  8      $550-^1.10  =  $500,  present  value  of  note  ; 
$480,  cash  offer  ; 

Ans,  $  20,  gain. 

Ex.  9.       $517.50-^1.035:=$  500 
$793.75-M.05f=:$  750 
$1326.47-f-1.105  =  $1200.426-f- 


$2450.426  +  ,  entire pres.  worth 
$2637.72-$2450.426  +  =:$187,29-f,^/^s. 

Ex.  10.  $.10  X  If  r=$y^^,int.  of  $1  for  10  mo.  at  10  per  cent 
$130^1y'2  =::$120  ;  $130  — $120  =  $10.00,  discount. 
$130  X  yV  =$10,831,  interest. 

Ans,  $.83^. 


PROMISCUOUS    EXAMPLES    IN    PERCENTAGE. 

(Pag^  260.) 
Ex.  1.     .02  4-  .25  =  . 27  gain  per  cent,  on  cost. 

V'  ^  fo  0  =^tV  <^ents,  selling  price  of  what  remains 
of  every  pound,  after  transportation  ; 

Ex.  2.     $200  X  .40  =  $80  gain  on  one  ; 

$200  X  .20     $40  loss  on  the  other; 

Ans,  $40. 


106  PEKCENTAGB. 

Ex.  3,     $425-^1.03=$412.62+  cash  value  of  sale; 
$425  — $25  =$400.00  cost ; 

Ans,  $1^.624,  profit. 

Ex.4      $.13-f-1.04  =  $.125;  $.13— $.125=$.005  ; 
$.005-r-$.125=.04,  ^?i5. 

Ex.  5.    $150-^-1.25=$120  costof  one; 

$150^    15=  200  cost  of  the  other; 

$320  cost  of  both; 
$320  — $300  =  $20,  Ans. 

Ex.  6.     Amt.  of  $1  for  2  yr.  8  mo.  at  9  per  cent.,  $1.24  ; 
jiyji  X  |«  J  X  f =$3750,  Ans. 

Ex.  1.     1.00  — .07  =  14f  years,  Ans. 

Ex.  8.     3  yr.  4  mo.=r3i  yr. ;    .121  x  3i=:.4is.,  whole  nito 
of  gain;  $5000-r-.41|  =  $12000,  capital ; 
$12000  X  f  =  $7500,  ^'s,  ) 
$12000  X  f —  $4500,  B's,  J 

Ex.  9.     $800  X  .15  =  $120,  gain  on  groceries ; 
500  X  .20:=   100,  loss  on  dry  goods  ; 

Whole  gain  $20,  Ans. 

Ex.10.    1.00-.08i=.91|; 

$1100h-.91|  =  $1200,  Ans. 

Ex.  II.     $667-f-1.04=$C41.346  +,  cash  value  of  goods ; 

600x1.06=  636.^       pres.  valuation  of  goods , 

True  gain,       $5,346  +  ,  Ans. 

Ex.  12      $18xi=$6,  profits;  $18— $6=$12,  cost; 
$6-7-12  =  50  per  cent.,  profit,  Ans, 

Ex.  13.  If  f  sell  for  f  of  cost,  the  whole  would  sell  for  f  x  | 
>f  the  cost,  which  is  1]|  times  cost.  Hence  JJ  =  .40f,  is  the 
S|;ain  per  cent. 


PROMISCUOUS   EXAMPLE?  107 

Ex.  14.     $1.30,  received  for  lumber  originally  wortli  $1.00  ; 
$1.06|,  valuation  of  ditto,  after  16  mo.  int.  accrues  ; 

■     $.231  gain  on  $1.06|; 

$.23iH-$1.0G|=.21|,  Ans. 

Ex.15.     1-1  =  1;  ix^^i,Ans. 

Ex.  16.     $1,121  x728  =  $819,  expended  in  wheat; 

.60  X  .30  X  .50=1.09  ;  $819^.09  =:$9100,in  bank; 
1.00  — .60^.40;  $9100x.40=:$3640,  Ans. 

Ex.  17.  6x5=i30sq.  yd.,  area  to  be  covered  ;  4  per  cent. 
IS  2^,  and  5  per  cent,  is  ^V  ;  hence  every  yard  purchased  will 
be,  after  shrinking,  ||  of  1  yd.  long,  and  if  off  yd.==|-J  yd. 
wide,  and  will  contain  ||  x  f  J  sq.  yd.  Therefore,  as  many 
yards  must  be  purchased  as  |i  x  f  i  is  contained  times  in  30. 

Ex.  18.  1.00  per  cent.=B's  money; 
1.28        "       =A's      " 


.28-4-1.28-. 211,  Ans, 

Kx.  19.  $1200-T-.12:=$10000,  f  of  his  capital; 
$10000-4-|=$25000,  whole  capital; 
$25000  X  f  X  yi^=$750,  loss  on  |  of  the  capital ; 
$1200— $750  =  $450,  gain. 

Ex.  20.  Amt.  of  $1  for  3  yr.  9  mo.  at  10  per  cent.,  $1,375  J 
$1933.25^-1. 375=:$1406=i  of  C's  money. 
$1406x2=±$2812,  C's, 


$2812 


X2=±$2812,  C's,  ) 
x#=$4218,D's,  )         • 


108  PERCENTAGE. 

BANKING. 
(336^   page   264). 

Ex.  1.  Int.  of  $450  at  6  per  cent.,  for  63  da.=$4.725    lisc't  : 

$450— $4,725=1445.275,  proceeds. 

Ex.  2.  Int.  of$368at7percent.,for93da.  =  $6.654+,disc't; 
$368  — $6,654+  =$361,345  +  ,  proceeds,  Ans, 

Ex.3.  Int.  of  $475.50  at  5  per  cent.,  for  63  da.=$4.16  +  , 
discount;  $475.50— $4.16  + =$471.33  +  ,  proceeds. 

Ex.  4.  Int.  of  $10000  at  6  per  cent.,  for  93  da.=$155,  disc't; 
$10000— $155  =  $9845,  proceeds,  Ans. 

Ex.  5.  Proceds  of  the  note,  disc'ted  at  6  per  cent.,  $247,375  ; 
$247.375— $240=$7.375,  Ans, 

Ex.  6.  Int.  of  $360.76  at  6  per  cent.,  for  93  da. =$5,591  +, 
disc't;    $360.76— $5.591 +  =$355,168+,  proceeds. 

Ex.  7.  Proceeds  of  the  note,  $529.2355  ; 

$530  — $529.2355  =  $.7645,  Ans. 

Ex.  8.  From  Mar.  2  to  Apr.  7  is  36  da.,  term  of  discount; 
Int.  of  $500,  at  6  per  cent.,  for  36  di.,  is  $3.00,  disc't ; 
$500-$3.00  =  $497,  proceeds. 

Ex.  9.  From  Nov.  15  to  Dec.  15  is  30  da   term  of  discount ; 
Amt.  of  $750  on   interest  for  6    no.  3  da.  at  0  pel 

cent,  is  $772,875  ; 
Bank  discount  of  $772,875  for  30  aa.,  x\  iO  km^  g^^av 
is  $6,440+  ;  $772.875— $r.44U^  ^  -^ 
$766,434+,  proceeds 


EXCHANGE.  109 


(337,  page   2G6.) 

Ex.  2.  $680-^.9895  — $687,215+,  Ans. 

Ex.  3.  $1000-^.9870|  =  $1013.085+,  Ans, 

Ex.  4.  $o00-+.9935f  =:$503.22+,  Ans, 

Ex.  5  $1256+- .96441=11302.341  +,  Ans. 


EXCHANGE. 

(350,  page   269.) 

Ex.  3.     $1000  X  1.03  =  $1030,  Ans. 
Ex.  4.     $400  X  1.0075  :=:r$403,  Ans. 

Ex.  5.     8530  X  1.0275rrr$544.575,  cost  of  draft ; 

20.  transportation ; 

Ans.  $564,575. 

Ex.  6.     $1— $.01225 =$.98775,  proceeds  of  $1  at  b'k  disc'tj. 
Add  .02         premium ; 

$1.00775,  cost  of  exchange  for  $1  ; 
$800  X  1.00775 =$806.20,  Ans. 

Fa.  7.     $1— $.0055  =  $.9945 

Subtract  .015      discount; 

$.9795,  cost  of  exchange  for  $1  ; 
$420  X. 9795  =$411.39,  Ans. 

Ex.  8.     $1  — $.0225=$.9775;  320  x  $10=$3200; 
$3200x.9775  =  $3128,  draft; 

312,  transportation  ; 
400,  gain ; 

To  be  sold  for     $3840 

$3840^320=$12,  Ans. 


110  PERCENTAGE. 

(351,  page  271.) 
Ex.  2.     $243.60^1.015=1240,  Ans. 

Ex.  3.     $79.20-^.99=:$80,  Ans. 

Ex.  4.     $1— $.0105  =  $  .9895 
Add  .02 


$1.0095,  draft  for  $1; 
$282.66^1.0095  =$280,  An€. 


Ex.  5.     $1— .0055=$.9945 
Subtract  .0125 

$.982,   draft  for  $1  ; 
$240-^.982  =  $244.399  +  ,  draft  hot.  for  $240; 
1240  — ($240  X. 005)=   238.80         current  money  for  $240  ; 

Ans.  $     5.599  +  ,  saved. 


Ex.  6.  $1.00000 

.01225,  bank  disc't  at  7  per  cent.  (63  da^ 

$  .98775 

.0075    premium. 

$3C00-h     .99525  =  $3617.181  +,  draft  required. 
$3600^   1.0075   =$3573.200  +  ,  draft  sent. 
Add  int.  for  60  da.  $     35.732+    (at  6  per  cent.) 

Amt.  at  time  req'd,    $3608.932  + 
$3617.181 -$3608.932  =$8.24+,  loss,  Ans. 


EQUATION    OF   PAYMENTS.  Ill 


EQUATION  OF  PAYMENTS. 

(356,  page  273.) 

Ex.  2.  $700x20  =  114000 
400x30=  12000 
700x40=   28000 


$1800  $54000 

54000^1800  =  30  da.  Average  credit 
Sept.  25-f30  da.  =  Oct.  25,  equated  time. 

Ex.  3.  $250x4  =  $1000 
750x2=  1500 
500x7=   3500 


$1500  $6000 

6000 -T- 15 00 =4  mo.  average  credit, 
July  1+4  mo. = Nov.  1,  Ans, 

Ex.  4.               $1x0=$  0 

2x1=  2 

3x2=  6 

4x3=  12 

5x4=  20 

6x5=  30 

7x6=  42 


$28  $112 

112-r-28=4  da. 
Monday -I- 4  da. = Friday,  Ans, 

Ex.  5  $650  X    4=$2600 

725  X    8=  5800 
500x12=   6000 


$1875     $14400 

14400^1875  =  7.68  mo.=7  mo.  20  da. 

Jan.  1  +  7  mo.  20  da. = Aug.  21,  Ans, 


112 


AVERAGINa  ACCOUNTS. 


Ex.  2. 


£x.  3. 


(362,  page  276. 


Due. 

da. 

Items. 

Prod. 

Jan.         1 

16 

Feb.        4 

March     3 

0 
15 
34 
61 

150 
200 
100 
160 

3000 

3400 
9760 

610 

16160 

16160^610  =  26  da. 

Jan.  l-}-26  da. = Jan.  27,  Ans. 


Due. 

da. 

Items. 

Prod. 

March     1 
April      4 
Aug.     18 
Aug.       8 

34 
170 
160 

300 

240 
100 
400 

8160 
17000 
64000 

1040 

89160 

89160  —  1040  =  86  da. 

March  1  +  86  da.=May  26,  Ans. 


Ex.  4. 


Due. 

da. 

Items. 

Prod. 

June       1 

600 

"        12 

11 

300 

3300 

"        15 

14 

832 

11648 

"        25 

24 

760 

18240 

30 

29 

750 

21750 

8242   , 

54938 

54938-^3242  =  17  da. 

June  1  +  17  da.=June  18,  Ans. 


AVERAGING    ACCOUNTS. 


118 


Ex.  5. 


Due. 

da. 

Items. 

Prod. 

Jan.       16 
Feb.      20 
March     4 
April    24 

35 

48 
99 

536.78 
425.36 
259.25 
786.36 

14887.60 
12444.00 

77849.64 

2007.75 

105181.24 

j  105181.24-^-2007.75  =  52  da. 
^^'  \  Jan.  16,  1856  +  52  da.=March  8,  1856. 


Ex.6. 


Due. 

da. 

Items. 

Prod. 

April      1 

"        28 

June      15 

27 
75 

420 

135 

1800 

3645 
135000 

2355 

138645 

A71S. 


(  139650-^2355  =  59  da. 
(  Apr.  1  +  59  da.=May  30. 


Ex.  2. 


(36 3,  page  278.) 


Br. 

Or, 

Due. 

da. 

Items. 

Prod. 

Due. 

da. 

Items. 

Prod. 

Jan.      1 

Feb.     4 

*'      20 

34 
50 

448 
364 
232 

12376 
11600 

Jan.    20 

Feb.    16 

"      25 

19 
46 
55 

560 
264 
900 

10640 
12144 
49500 

104423976| 

172472284 

1044'23976| 

Balances 

68048308 

Ans, 


48308-^-680  =  71  da. 
Jan.  1+71  da.=March  13 


114 


RATIO. 


Ex.3. 


Dr. 

Cr. 

DnSf 

da. 

Items. 

Prod. 

Due. 

da. 

Items. 

Prod. 

Apr.    1 
June  12 
Sept.  3 
Oct.     4 

0 

72 

155 

186 

145.86 
37.48 
12.25 
66.48 

2698.56 

1898.75 

12365.28 

May  11 
July  12 
Oct.  12 

40 

102 
194 

11.00 
15.00 
82.00 

440.001 

1530.00 

15908.00 

262.07 
108.00 

154.07 
— 

16962.59 

108.00 

17878.00 
16962.55 

Bal  acct. 

Bal.  Prod. 

915.41 

(  915.41-f-154.07  =  6  da. 
'  I  Apr.  1  —  6  da.=:Marcli  26,  1858. 


RATIO. 


Ex.  3. 

3V  =  1,  Ans. 

Ex.  5. 

^  =  5l,Ans. 

Ex.  7. 

\*~Xj%  =  l2,  Ans, 

Ex.9. 

ixi=i,Ans. 

Ex.  11. 

3-0x-^-z=5,  Ans. 

(379^  page  281.) 

Ex.  2.  2\={,  Ans, 

Ex.4.  ^f^=1,Ans. 

Ex.  6.  2J.x}={i,Ans, 

Ex.8,  ^f  =  41,  Ans. 

Ex.  10.  -j4_x|=:|,  ,Ans, 

Ex.  12.  3  gal.  =  24  pt. ;  2  qt.  1  pt.  =  5  pt. ; 

5-^24  =  2^7  ■^^^' 
Ex.  13.  8  s.  6  d.=:8.5  s. ;  |;f =|f  =  1||,  Ans. 
Ex.  14.  -/I^/Zo  =tV,  ^ris. 

Ex.  15.  19  lb.  5  oz.  8  pwt.=4668  pwt. ;  25  lbs.  11  oz.  4 
pwt.  =  6224  pwt.  |||i  =  li,  Ans. 

Ex.  16.  ||=f,  Ans.  Ex.  17.  f  x  l  =  j\y  Ans. 

Ex.  18.  2^3  x^^rzz^,  Ans.         Ex.  19.  16-r-2f  ==7,  Ans. 


SIMPLE   PROPORTION.  115 

Ex.  20.  14.5  X  3  =  43.5,  Ans.     Ex.  21.  J  X  f =|  =  li,  Ans. 
Ex.  22.  ^xiz=j\,Ans. 


PROPORTION. 

(388,  page  283.) 

„     ,       48x50     ,    ^     .  ^     ^      70x3     ^     . 

Ex.  1. =120,  Ans,      Ex.  2.     ■— —  =  5,  Ans, 

20  '  42  ' 

Ex.  3.  =6,  Ans.  Ex.  4.     -^=12,  Ans. 

_     ^      201.75x48  yd.     ^^^     ,      . 

Ex.  6.     3  lb.  12  oz.  =  60  oz. 
10.50x60  oz. 


3.50 


=  180  oz.  =  ll  lb.  4  oz.,  ^n5. 


Ex.  Y.     8  bu.  2  pk.=34  pL;  1Q  bu.  2  pk.:   ?06  pk. 

$38.25x34     ^,     ,     , 

=$4.25,  ^n5. 


306 


Ex.  8.     V  X  ^^  X  ^1  ^= Y-=8i,  ^Tis. 

Ex.  9.     V  X  I  X  5  =  7,  ^?25.        Ex.  10.^  X I  X  f =|,  ^ns. 


SIMPLE  PROPORTION. 
(3995  page  287.) 


Ex  1.    48  Cd.  :  20  Cd.  : :  $120  :  (    ) 

,     ,     $120x20     ,^     . 
(     )=^-jg =$50,  ^/w 

Or.  $120xf?  =  $50,  Ans, 


116  PKOPORTION. 

Ex.  2.     6  bu.  ;  75  bu.  :  :  $4.75  :  (     ) 

,     ,     $4.75x75     ^  ^    ^,     , 
(     )  = =$59,371,  ^W5. 

Or,  $4.75  X  V  =$59.37^,  Ans. 

Ex.  3.     $3i  :  $50  : :  8  yd.  :  (     ) 

,     ,     50x8  yd.     ^^^,      ,      . 

Ex.  4.     12  :  20  : :  42  bu.  :  (     ) 

,     ,     42  bu.  X  20     .  ^  ,  . 

(     )=z -^ :^10hu.,  Ans. 

Or,  42  bu.  X  ^:=^10  t>u.,  Ans. 

Ex.  5.     $.75  :  $9.00  : :  7  lb.  :  (     ) 

,     ,     900  X  7  lb.     ^^  ,^     , 
(     )=-Y^ ^84lb.,  ^7Z5. 

Ex.  6.     3  lb.  12  07.  :  11  lb.  4  oz.  : :   $3.50  :  (    ) 

60  oz.  :  180  oz.  ::  $3.50  :  (     ) 

,     ,      $3.50x180     ^         -^     , 
(     )  =  - =$10.50,  Ans. 

Ex.  7.     1  ft.  6  in.  :  75  ft.  : :  3  ft.  8  in.  :  (     ) 
lift.  :  75  ft.  ::  3|  :  (     ) 

(     )  =  Y  x-y  ft.  x|==183J  ft.  =  183  ft.  4  in.,  ^n*. 

Ex.  8.     $2.75x  VV  =$19-^^7 )  ^^^^• 

Ex.  9.     $13.32  :  $51.06  ::  12  bu.  :  (     ) 

,     ,     51.06  x  12  bu.      ^^, 

(     )  — —  46  bu.,  Ans. 

^     '  13.32  ' 

Ex.   10.   15hhd.i=945  gal. 

945  gal  :  28.5  gal.  :  :  $236.25  :  (     ) 
,     ,      $236.25x28.5      .   ^^,      . 
(     ^" ^45" -'^.ISi,  ^/^s. 


SIMPLE    PROPORTION.  '        117 

Ex.  11.  6  mo.  :  11  mo.  : :  1  bbl.  :  (    ) 

,     .     11  xY  bbl.     ,^,  ,, ,      . 
(     )=~ =l2^hhl,Ans. 

Ex.  12.  5  £  12  s.  :  44  £  16  s.  : :  9  yd.  :  (     ) 

,     .      9  yd.  X  896     .^     ,       . 
(     )=-^-u^ =  72  yd.,  ^715. 

Ex.  13.  $3100  X  J7//^=$310,  Ans. 

Ex.  14.  100  lbs.  cofFee=100  X  1  =  160  lbs.  sugar; 

2  :  160  : :  $.25  :  (     ) 

,     ,     $.25x160     ^^^     , 
(     )= =$20,^715. 

Ex.  15.     13°  10'  SO' :  360°  :  :  1  da. :  {    ) 
47435''  :  1296000"  : :  1  da.  :  (     ) 
(     )r=J-f i| jiiL  da.  =  27  da.  7  h.  43  min.  6.06  +  sec,  Ans. 

Ex.  16.     8J  :  13^  : :  $4.20  :  (    ) 

(     )  =$4-f-o  X  V-  X  3V=$6.48,  Ans. 

Ex.  17.     6J  d.  :  10  £  6  s.  8  d.  : :  If  yd.  :  (     ) 

(     )=2.4_8jL  X  J  yd.  X  2T  =  694|  yds.,  Ans. 

Ex.  18.     121  cwt.  :  48f  cwt.  : :  $421  :  (     ) 

(     )  =  $i|5.xi|ix  22j=$163.50  +  ,  Ans. 

Ex.19.    $lf  :  $317.23  ::  8|  lb.  :  (     ) 

(     )=:317.23  X  8.4  lb.  X  4  =  1522.7+  lb.= 
15  cwt.  22.7+  lb.,  Ans. 

Ex.  20.     $1561  :  $95.75  : :  15f  bu.  :  (     ) 

(     )  =  ^J^■LS-x^^  bu.  615=9.575  bu.= 

9  bu.  2  pk.  2 1  qt.,    Ans. 

HJx.  21.     I  bar.  :  1  bar.  : :  $^|  :  (     ) 

(     )=$^  xixi  =  $j^j,  Ans. 

K.  P  6 


118 

Ex.  22. 

Ex.  23. 
Ex.  24. 

Ex.  25. 


PROPORTION 

4  rd.  :  llf  rd.  ::  f  A.  :  (     ) 

(     )=:}  A.  X  V-  X  i  =  2^\  A.=2  A.  28  rd.,  ins 

13cwt.  :  12  cwt.  ::  $421  :  (     ) 


16  oz.  :  12  oz.  : 
3x12 


(  y- 


16 


:  $28  :  (     ) 

=$21,  ^/i5. 


16  oz.— 1411  Q2.=ly5_  oz.,  cheat  in  16  oz, 

16  oz.  :  l/g  oz.  : :  $30  :  (     ) 

(     )=:$3jP  X  2.1  XyV=$HI  ==$2.46 +  ,  Ant 

Ex.  26.     1  yr.  6  mo.  :  3  yr.  9  mo.  : :  $750  :  (     ) 
,     ,     $750x45     ^     ^ 
18 

Ex.  27.     10  mo.  x  V/r-^^^  ^o->  ^^^• 

Ex.  28.     $25  :  $30^  : :  $28  :  (     ) 

(     )=$2J-  X  V-  X  2V  =  $34.16,  Ans. 

Ex.  29.     1  yr.  4  mo.  =  li  =  |  yr. ; 

i  yr.  X|J|  =  2J  yr.  =  2  yr.  9  mo.  10  da.  Am 


COMPOUND  PROPORTION. 

\ 

(401 5  page  292.) 

.  1.     16 

50  J 

/o[-^^«  =  (    )         - 

128 
5 

(  ) 

90 

(     )  =  72  bii.,  Am. 


COMPOUND   PROPORTION. 


119 


Ex.  2.     3 
12 


h<4 


120 


; 360    120 
10 

(     ) 


300 
12 


54 


(     )  =  10f  days,  Ans. 


Ex.3. 


6 

10 


(     ) 


34 
20 
15 


(     )  =  170  yards,  Ans. 


Ex.  4 


450        (     ) 

12  V    :      9 
12  )  8 


1  :  1 


8 

9 

(     ) 


12 

12 
450 


(     )=900,  Jtis. 


Ex.  5.     1200  ; 


Ex.  6. 


Ex.7. 


wi 


500  :  960 


m-- 


4 

^ 

8J 


■■v\-- 


48 


6|:(    ) 


500 

960 

1 

8 

4 

5 

(     ) 

1200 

1 

23040 

(     )=3291^,  An9 


36 
12 
12 

(     ) 


48 

9 

9 

8 


(     )=:6  men,  Ans, 


V-xVx*xYxixi>< 7V=¥i^  =  40tt»  ^^*' 


120  PROPORTIDN. 

Ex.8.     41  )         4|   } 

6  ^  :     9     V  :  :  540  :  GOO 

20  )      (     )  ) 


Ex.  9.     2^ 


If 


[  :^;|  j    ::$3.27i:(     ) 


.6.75      73     3     2     5     J478.25     ^_  ^^ 


Ex.  10.      5 
6 


[   •  ^  12    [   •  •  ^^-^  •  ^^'^•^ 

417.6x5x6      ^^ 
— ^ttt: — 7^ —  =  20  men,  Ans, 
52.2x12  ' 


Ex.  11.  6    )  9    )         22.5      )       45 

2.5    y  :  (     )    [  ::  17.3      [  :  34.6 
12.3    )         8.2    )         10.25    )       12.3 

6x2^x12^3x^5  X  34.6  X  12.3  _ 
9x8.2x22.5x17.3x10.25    *"^^  ^^^^'  ^'''' 


Ex.  12.  54  J  75 
24J  f  :  (  ) 
121  )         lOJ 

V-  X  V  X  V  X  T  J  X  /t=21  days,  Ans. 

Ex.  13.     24  ^      217     J         33J  )      23i 

189  [  :       51  V  : :     5| 


U)      {    )    )  31)        21 

Vx^l^XVxVxy  XlX2lTXi^TXHjX^YXf  =  16, 


^7lS. 


PARTNERSHIP. 


121 


Ex.2, 


$  8000 
12000 
20000 

$40000 


PARTNERSHIP. 
(407,  page  295.) 
TVoW=i»    A's  fraction. 

±2  0.0.0. _3_     TJ'q  U 

40000  10>    -^^ 

2.00.00.  —  i       n\         a 


-$336,   A's;    $1680  x  tV=$504,    B's 


$1680xi==:$840,  C's. 
Ex.  3.  $1200+$1000  +  $600=::$2800; 


$2800 

:  $1200  : 

'  $224 

:  ( 

)=:$96,  A's  share; 

$2800 

:  $1000  : : 

$224 

'{ 

)  =  $80,  B's   " 

$2800 

:  $  600  : , 

$224 

( 

)=$48,  C's   " 

Kx.  4.  $20000  :  $13654  ::  $3060  :  (  )=$2089.062 
$20000  :  $13654  ::  $1530  :  (  )=$1044.531 


Ex.  5.  16  +  24  +  28  +  36  =  104 
$13XyV4=^2,  A  pays; 


'x/oV=^3,  B  pays; 


$13  X  tVt  ==^3.50,  C  pays ;  $13  x  j\\=$^,50, D pays. 

Ex.6.     14  +  6  +  12  =  32  shares. 

$2240  X  i|=:$980.  Captain's  share  ; 
$2240  X  3\=:$420,  Mate's  share  ; 
$2240  X  i|=$840,  divided  among  the  sailors  ^ 
$840-r-12=$'70,  each  sailor's  share. 


fix.) 


$3475.60— $2512=$963.60,  lost  to  the  owner*  ; 
$963.60  X  i=$120.45,  A's  \ 
$963.60  X  1  =  $240.90,  B's  i  Jns. 
$963.60  X  f  =  $602.25,  C's  ) 

Proof    1  =$963.60 


122  PARTNERSHIP. 

•    Ex  8.  6,  C's  proportional  share. 

6  f  4=10;  10x1=  2,  E's  **  ** 

6  +  4  +  2  =  12 

$2571.24  X  ^2  =$1285.62,  C's  ;  \ 
$2571.24  X  yV=$  857.08,  D's ;  V  Ans. 
$2571.24  Xy2-=$  428.54,  E's;) 

Ex.  9      $7500-($2000  +  $2800.75  +  $1685.25)  =  $1014, 
D's  gain ; 

gain.  cap.  gain.  cap. 

$1014  :  $3042  ::  $2000    :  (  )=$6000   Am 
$1014  :  $3042  ::  $2800.75  ,  (  )=$8402.25,  B;  i  Am, 
$1014  :  $3042  ::  $1685.25  :  (  )  =  $5055.75,  C ;  ) 


(408,  page  297.) 

Ex.  2.     $250  X  6 =$1500,  B's  product ; 
275x8=   2200,  C's       " 
450x4=   1800,  D's      " 

$5500 
$825  X  If  =$225,  B's  share  of  gain ; 
825  X 11=   330,  C's      "      "      " 


825X;?-|=  270,  D's  " 

(( 

$1000  X  8=$  8000 

1600x10=  16000 

Ex.3.     $1000  X    8=$  8000  $1500  x    4  =  $  600C 

1200x14=   16800 

A's  product,  $24000  B's  product,  $22800 

$2l000  +  $22800  =  $46800,  sum  of  products. 
$46800  :  $24000  ::  $1394.64  :  (     )  =$715.20,  A's  gain  • 
$46800  :  $22800  ::  $1394.64  :  (     )=$679.44,  B's     " 


PARTNERSHIP. 


123 


Ex.  4.     4x5  days =20  days'  work  A  furnished ; 
3x6     "    =18     "         "      B        " 
6x4     "    =24     "    .     "      C        " 

62     "         "      all       " 
372  buslie]s-f-4  =  93  bushels  to  be  divided. 
62  :  20  ::  93  :  (     )=30  bu.,  A's ;  j 
62  :  18  ::  93  :  (     )  =  27    "     B's ;  [  Ans. 
62  :  24  ::  93  :  (     )=36    "     C's;  J 

Ex.  5. 

From  Jan.  1,  1856,  to  Apr.  1,  1858,  is  27  mo.,  Gallup's  time ; 

"     Mar.  1,  1856,  "  Apr.  1,  1858,  "  25    "     Decker's    " 

"     July  1,  1856,  "  Apr.  1,  1858,  "  21     "     Newman's" 

$3000  X  27  =  $81000,  Gallup's  product ; 

2000x25=   50000,  Decker's      " 

1800x21=  37800,  Newman's    " 

$168800,  sum  of  products. 
$4388.80  :  (     )  =  $2106,  Gallup's  gain; 
$4388.80  :  (     )  =$1300,  Decker's     '' 
$4388.80  :  (     ) =$982.80,  Newman's  ". 


$168800 
$168800 
$168800 


$81000 
$50000 
$37800 


Ex.  6.     $560—  8=$70,  A's  monthly  profit; 
$800^10=$80,  B's  "  " 

$150 
Since  the  gains  of  the  partners  are  proportional  to  their 
amounts  of  capital  when  the  times  are  equal,  we  have 
$150  :  $70  :  :  $5600  :  (     )  =$2613.331-,  A's  gain; 
$150  :  $80  :  :  $5600  :  (      )  =  $2986.6G|,  B's  gain. 

Ex.  7.  If  we  allow  2  parts  of  the  gain  to  A,  3  parts  to 
B,  and  4  parts  to  C,  |  of  A's  gain  will  be  equal  to  I  of  B's, 
and  to  J  of  C's,  and  the  proportion  of  the  shares  will  corres- 
Dond  to  the  conditions. 


124 


ANALYSIS. 


2+3+4=9 

$117  X  |=:$26,  A's  gain  . 
$ll7xf=$39,  B's     " 
|ll7x^riz$52,  C's     " 
If  we  now  divide  the  proportional  shares  of  the  gain,  2,  3,  4, 
by  the  respective  times,  3,  5,  7,  we  shall  obtain  the  piopor- 
tional  monthly  shares  of  the  gain,  which  must  be  in  the  sa.i:*e 
proportion  as  the  respective  shares  of  the  capital ; 

2-f-3  =  |,  A's  proportional  share  of  capital  • 
3-+5  =  f,  B's  "  "      " 

4-h7=4,  ^'s  "  "      "         " 

l"^! +  7=Tol»  ^^™  of  proportional  shares. 
:  (     )  =  $700,  A's  capital ; 
:(     )=$630,  B's        " 
:(     )=$600,  C's 


tf f  :  I  :  :    $1930 

19  3    .    3 
105*5 


HI:  4 


$1930 
$1930 


ANALTSia 


(412,  page  300.) 

Ex.  5.  We  multiply  the  number  of 
casks  by  the  number  of  pounds  per 
cask,  and  this  product  by  the  number  of 
pence  per  pound,  and  obtain  the  cost  in 
pence;  which,  divided  by  56,  the  number 
of  pence  in  a  dollar,  gives  $27,  answer. 


56 


3 

126 

4 


$27,    Ans, 


Ex.  6.     We  multiply  19  (pence)  by  28  7 

for  the  cost  of  the  butter  (in  pence),  and  j_^_ 

livide  by  1  times  12  (pence)  the  price  of  _£_ 
the  tea. 


28 
19 
"l9 


61,  Ans. 


ANALYSIS. 


125 


Ex  7.  10  s.  6d=126d.  The 
product  of  2,  72,  and  4  is  the  num- 
ber of  quarts.  Multiply  this  oy 
126,  the  selling  price  per  quart, 
and  divide  by  96,  to  reduce  the 
result  to  Decimal  currency,  and  we  $756  — 
obtain  $756.  Subtracting  the  cost 
we  obtain  the  profit. 


96 


2 

72 

4 

126 


$756 
=  $108,  An9. 


Ex.8.  2  s.  6d.=:30d.  Then  2  x  3  x 
7  X  30= cost  in  pence.  Divide  by  60  to 
reduce  to  Decimal  currency. 


60 


2 
3 

7 
30 


$21,  Ans. 


Ex.  9.  20  X  3  X  12r=value  of  the  ap- 
ples in  pence.  Divide  by  6  s.  3  d. 
(=75  d.)  to  find  the  number  of  days' 
work  to  be  given  in  exchange. 


20 

3 

75 

12 

5 

48 

9|,  Ans. 


Ex.  10. 


96 


160 
18 


Cost,  $30 
$42.66f- 


90 


160 
24 


sold  for  $42.66|. 
)=$12.66|,  ^/i5. 


Ex.  11.     431=  V;  10  s.  6  d.= 
8  s.  3  d.  =  99  d. 


:126d. 


Ex.  12.     9  s.  4  d.  =  112  d.  ; 

$1=96  d.,  Mich,  currency 


2 
99 

87 
126 

11 

609 

65y*i-     A718, 


96 


300 
112 


$350,  Am 


126 


ANALYSIS. 


128 

5 

90 


Ex.16.     Dividing  128  by  16,  we  ob- 
tain what  1   horse  Avill   consume  in  50         iq 
days ;  dividing  this  result  by  50,  we  ob-         50 
tain  what  1  horse  will  consume  in  1  day.  72,  Ans, 

Multiplying  by  5,  we  find  what  5   horses 
will  consume  in  1  day ;  and  multiplying  this  result  by  90,  we 
find  what  5  horses  will  consume  in  90  days. 


Ex.  11.  Divide  4|  (rr:Y)  ^7  l^i 
(=%')  to  find  what  amount  of  wood  $1 
will  buy  ;  then  multiply  by  24:^[=:\^)  to 
find  how  much  824J  will  buy. 

Ex.  18.  52  X  3  X  ^^=  the  money 
given  for  the  cloth.  Divide  this  result  by 
65,  the  number  of  yards,  to  find  the  price 
per  yard. 


Ex.  19.  A  shadow  of  1  foot  will  require  an  object  J  of  3 
feet  in  length  ;  and  a  shadow  of  46|  feet  will  require  an  object 
46  J  times  ^  of  3  feet  in  length ;  hence 

^xlx^^=28ieet,Ans. 


s 

14 

21 

2 

4 

99 

U.Ans. 

52 

3 

3 

100 

65 

Ans 

.  80  cents. 

Ex.  20. 
8  sheep  x  Y|=  60  sheep  for  1  mo.,  A's  use  of  the  pasture  ; 
12 
15     "     vfi2_inn     «       "        "     P.'a    "      "  « 


X4i=:   50 
X6|=100 

u 
u 

"  ,B's    " 
"  ,C's    " 

"  ,  total  " 

210 

(C 

Each  man  should  pay  such  part  of  the  whole  cost  as  his  use 
of  the  pasture  is  part  of  the  total  use ;  hence 

$63  X  2^=118,  A  must  pay  ; 
$63x^y=:$15,  B     "         " 
$63xif==|30,  C     "         " 


ANALYSIS. 


127 


Ex.  21.     1  bu.  oats    =1  dollars  ; 

1  bu.  rye      =-y  bu.  oats  =  Y  ^  i  <iollars  ; 
1  bu.  wheats Y  bu.  rye=Y  X  Y  xf  <iollais. 
If  we  divide  $30  by  the  price  of  1  bushel  of  wheat,  we  shall 
have  the  number  of  bushels  which  $30  will  buy  ;  hence 
V-  X  f  X  y\  X  tV=15  bu.  Ans. 


Ex.  22.  If  $480  gain  $84  in  any  time, 
Ic  gain  $21  in  the  same  time  will  require 
II  of  $480  ;  and  if  |i  of  $480  gain  $21 
in  30  mo.,  to  gain  the  same  amount  in  15 
mo.  will  require  f  |  of  |i  of  $480. 


84 
15 


480 
21 

30 


$240,  Ans, 


Ex.  23.  28  X  f  =  21  sq.  yd.,  contents  of  the  28  yd. ; 
21-h|  =  31i  yd.  of  that  which  is  |  yd.  wide. 


Ex.  24.  If  130  miles  require  3   days,    390 
miles  will  require  f  fj  of  3   days  ;  and  if  14    I^^ 

hours  a  day  require  3  days,  7  hours  a  day  will    

require  y-  of  3  days. 


3 

390 

14 


18, 


Ans. 


Ex.  25.  If  6  men  cut  45  cords  in  any  time,  8 
men  can  cut  |  of  45  cords  in  the  same  time  ;        ^ 
and  if  in  3  days  any  number  of  cords  be  cut,  in        _ 
9  days  there  will  be  cut  |  times  as  many  cords. 


45 

8 
9 


180, 


A71S, 


Ex.  26. 

A's  age  +  13's  age= 1  -f- 1|  =2^  times  A's ; 

C*s  age  =  2 yV  times  this  sum  —.5}     "        " 

And  the  sum  of  all  their  ages,  or  93  yr.  =  7^     "         *' 
Hence,  93^7f  =  12  yr.,  A's  age 

12x1^  =  18    "    B's     "      [Ans. 

12x5 J- =  63    "    C's     " 


128  ANALYSIS. 

Ex.  27. 

1  clay  ofC  =j%  da.  ofD.; 

1  day  of  B=  V  da.  of  C  =  ^  x  i%  ^^'  o^  ^^  i 
1  day  of  A^l  daofB  =f  x  V  x  A  ^a.  of  D; 
lience  5  days  of  A=^  X  |  X  Y  x  A  ^^-  ^f  D=8  da. 

of  D,  Ans. 

Ex.  33.  If  the  cost  of  12   oranges  and         O-        ^-     ^ 
10  lemons  is  54  cents,  the  cost  of  one  half  ,o     tq     54 

the  lot,  or  6  oranges  and  5  lemons,  will  be         ~^ r — -^ 

27  cents.  2     6 

But  the  cost  of  6  oranges  and  7  lemons     1  lemon =3  cts. 
is    33    cents.      And,   by   subtracting,   we    1  orange=2  cts. 
find  the  cost  of  2  lemons  to  be  6  cents, 
which  gives  the  cost  of  1  lemon  3  cents.     From  the  first  ex- 
pression, 6  oranges  and  21  cents  (equal  to  7  lemons)  is  equal 
to  33  cents ;  hence  6  oranges  are  worth  12  cents,  and  1  orange 
is  worth  2  cents. 

Ex.  34.  18x20x1000  =  the  whole 
number  of  ounces  of  provisions ;  and  since 
this  quantity  is  to  supply  1600  men  30 
days,  we  divide  by  30  to  find  the  daily  al- 
lowance for  the  army,  and  this  result  by  ^^*  2»  ^^* 
1600  to  find  the  daily  allowance  to  each 
man. 

Ex.  35.  If  we  add  6  bushels  to  the  smaller  bin,  there  will  be 
60  bushels  in  both  ;  but  as  the  larger  will  then  contain  2  time« 
as  many  bushels  as  the  smaller,  the  two  together  wi'l  contaiB 
three  times  the  number  in  the  smaller ;  hence 
3  times  the  smaller =60 
The  smaller =20 
The  larger  =20  x  2  =  40,  Ans, 


18 

30 

20 

1600 

1000 

15 


ANALYSIS. 


129 


Ex.  36.  We  take  the  difference  of  two  numl)<)r«  l>rom  tlve 
greater  to  find  the  less.  The  greater  diminished  by  i  of  the 
greater  equals  the  less,  which  must  be  |  of  the  greater.  And 
if  the  less  be  |  of  the  greater,  their  sum,  20,  is  1|  times  tb<? 
greater.     Hence  we  have 


20- 


^3 


:12,  the  greater,  Ans. 


Ex.  37.  1  day  of  C 
1  day  of  A: 
6  days  of  A 
6  weeks  of  A 


f  da, 


=  f  da.  of  B  ; 

ofC=:       fxf  da.  of  B;  hence 
=  6  X  f  X  f  da.  of  B  ;  and, 
=6x1x1  wk.  ofB. 
6  X  f  X  f  wk.  =  lli  wk.,  Ans, 


Ex.  38.  36  X  1J=45  sq.  yd.  to  be  lined. 
45  yd.-f-J  =  60  yds..  Arts. 


Ex.  39.  80  X  31  X  96  =  value  of  the 
broadcloth,  in  pence  ;  104  x  10= value  of 
one  sack  of  coffee,  in  pence  ;  and  to  obtain 
the  number  of  sacks  we  divide  the  former 
product  by  the  latter. 


80 

4 

13 

104 

96 

10 

24,  Ans. 


Ex.  40.  If  the  time  past  since  noon  is  equal  to  J  of  the  timo 
to  midnight,  both  intervals,  or  12  hours,  must  be  1;^  times  the 
time  to  midnight ;  hence 

12  h.-^H     =10  h.  to  midnight. 

12  h.— 10  h.=  2  h.  p.  M.,  Ans. 


Ex.  41    She  bought  one  half  for  i  cent  apiece ; 
And  the  other  half  for  i  cent  apiece. 
(2  +  3)"^^  — t\»  average  buying  price  ; 
3-T-5  =1,    selling  price. 

A        —ih  g^^^  o^  ^°®  peach. 

55-r  11  =  300,"  ^71.9. 


I~r^ 


6* 


130  ANALYSIS. 

Ex.  42.   A  can  build  the  boat  in  18  x  10  =  180  hours ; 
B    "       "       "      "      "     9x    8=   72     " 
A    "       "     yI  0  of  the  boat  in  an  hour  ; 

BU  U  1  ((  u  u  u 

A  and  B  can  build  TsT  +  TV^se  o    ^^  ^^^  ^^^^  '^ 

an  hour ; 
A  and  B  can  build  3  J^  x  6=/^-  of  the  boat  in  a  day 

of  6  hours. 
It  will,  therefore,  require  as  many  days  as  7  is  con- 
tained times  in  60  ;  hence 
60-^7=^84  days,  Ans, 
Ex.  43.  He  spent  at  first  ^,  and  he  had  ^  left.     He  then 
spent  A  of  this  ^,  and  he  had  |  of  this  ^  left ;  hence  |  of  ir= 
^  of  his  money,  which  is  $10,  and  the  whole  is  $30,  Ans. 

Ex.  44.  4  times  the  work  will  require  4  times  as  many  men, 
and  1  of  the  time  will  require  5  times  as  many  men  ;  hence 
30x4x5  =  600,  Ans, 
Ex.  45.  If  $3.25. buy  16.25  lb.,  $1.25  will  buy  Jf  |  of  16.26 
lb. ;  hence 

16.25  lb.xi|}  =  6.25  =  6i  lb.,  Ans, 

Ex.  46.  On  every  idle  day  he  lost  the  forfeit,  $1,  and  his 
wages,  $2.50,  which  together  amount  to  $3.50.     Had  he  la- 
bored every  day,  he  would  have  received  $2.50  x  20  =  $50. 
$50  — $43  =  $7,  what  he  lost  by  being  idle  ; 
and  $7-f-$3.50  =2,  the  number  of  idle  days.     Hence 
20  —  2  =  18,  the  number  of  days  he  labored,  Ans, 
Et.  47.  A,  B,  and  C  perform  ^j  in  an  hour 

AandB  "  yV     "        " 

Hence,  C  performs  ^j — ^=zj\  "  " 
Again,        A,  B,  and  C  perform  ,'^     "         " 

AandC  '•  yV     "         " 

Hence,  B  performs  tj—tV^sV  "  " 
Therefore,  B  and  C  perform  ^V  + 3V  ==tJ4  "  ** 
And  in  9^  hours  they  will  perform  yj^  x  Y  =  J,  Atvs. 


ALLIGATION.  V61 

ALLiaATION. 
(414,  page  305.) 


Ex.  2.     61.00  Xl2  =  $12.00 
1.50  X    5=     1,50 
3 


20    )$19.50 


$  .975,  Ans. 


Ex.  3.     $1.25x52  =  165 
13 


65       65 

Mixture,  $1  per  gal. 

65  X  32  X  $.06J=$130,  receipts  ;  $130— $65=$65,  gain, 

Ex.  4.       8x10=  80  Ex.6.     12    x    1^=  90 

9x12  =  108  101 X    8   =   84 

11x16  =  176  11    X    9   =   99 

-     —  10    xl0_i  =  105 

38x10  =  380  35       378 

A       "TT      4.  378x11=567 

Ans,     16  cents.  ,  ^ 

Ans.  567-^35=16icts. 


Ex.6. 

50  X    4=200  lbs.;  $.13  x200=$26.00 

4Cx  10  =  400  lbs.;  .10x400=  40.00 

25x24  =  600  lbs.;  .07x600       42.00 

1200)$1Q8.00 

$.09  average  cost  per  lb 
$.095  — $.09  =  $.005  ;  1200  x  $.005 =$6,  Ans. 


132 


ALLIGATION. 


(416,  page  310.) 


10 

i 

1 

1 

11 

1 

2 

2 

14 

i 

4 

1 

1 

2 

Ex.  3.     12 


Ans,  1  lb.  at  10,  and  2  lbs.  at  11  and  14  cents. 


Ex.  4.     90 


(    120 


9  0 
_!_ 
30 


Ans.  1  gal.  of  water  to  3  gal.  of  wine. 


Ex.5.     21 5  i 


'200 

tV 

3 

3 

250 

ii 

1 

1 

300 

ii 

1 

1 

400 

rh 

5 

5 

Ans,  5  of  the  1st  kind,  1  of  each  of  the  2d  and  3d,  and  3  of 
the  4th. 


Ex.  6.     90  < 


fso 

84 
87 
94 


6 

6 

i 

6 

6 

i 

4 

4 

i 

3 

3 

\ 

10 

6 

16 

-4w5.  6  of  the  first  2  kinds,  4  of  the  third,  3  of  the  fourth, 
and  16  of  the  fifth. 


(ilT,  page  311.) 


Ex.  2.     80  < 


'40 

tV 

2 

2      20 

60 

1 
US 

2 

2      20 

75 

i 

2 

2      20 

90 

1 

TIT 

tV 

tV 

8 

4 

1 

13    130 

Ans,  20  lbs.  of  each  of  the  first  three  kinds,  aL  s  130  Ib«.  lA 
tlje  fourth. 


Ex.  3.     4  ] 


2 

-i 

1 

1 

3 

1 

1 

1 

6 

1 

1 

2 

1 

3 

24 
24 
72 


Ans.  24  at  $3,  and  72  at  i 


ALLIGATION. 


183 


7V 

4 

4 

60 

sV 

4 

4 

60 

tV 

1 

40 

9 

3 

12 

180 

Ex.  4.     90  hJ  60 

130 
Ans.  60  gallons  each  of  alcohol  and  water. 


Ex.  5.     40 


35  1   } 


50 


TIT 


150 
15 


Ans,  150. 


Ex.  6. 


f  10    I 


f 

4 

4 

8 

80 

6 

6 

60 

f 

2 

2 

20 

Ans.  60  lbs.  at  Si  cts.  and  20  lbs.  at  10  cts. 


(418.   page   312.) 


Ex.  2.     14  < 


(9 

i 

6 

6 

60 

12 

1 

2 

4 

4 

40 

18 

1 
4 

2 

2 

20 

20 

i 

5 

5 
IT 

50 
170 

Ans,  60  at  9  s.,  40  at  12  s. ;  20  at  18  s.,  and  50  at  20  a 
Ex.  3.     22  < 

Ans,  6  ounces  each  of  the  first  three,  and  33  ounces  of  the  la*  t 
Ex.  4.     $l78.50-4-210=$.85,  average  price 


ri6 

i 

2 

2 

6 

18 

i 

2 

2 

6 

21 

1 

2 

2 

6 

24 

i 

i 

i 

6 

4 

1 

11 
17 

33 
51 

'    50 

1 

3  J 

13 

1 

13 

78 

85  H 

:  ^5 

tV 

13 

13 

78 

( 

[  150 

eV 

aV 

7 

2 

9 

54 

'    1 

35 

210 

Ans,  78  bu  each  of  oats  and  corn,  and  54  bu.  of  wheat 


184 

EVOLUTION. 

(  45 

4 

\ 

1 

2 

3 

3600 

Ex.  5. 

48  . 

51 

\ 

1 

1 

1200 

J54 

I 

6 

1 

1 

1200 

1 

5 

6000 

Arts.  A  3600  bu. ;  B  and  C  each  1200  bu. 
Ex.  6.     $84 -T- 5 6 =$1.50,  average  daily  wages. 


150^ 


f  60 

rh 

6 

6 

24 

15 

tV 

1 

1 

4 

115 

Vt 

3 

3 

12 

300 

Th 

4 

4 

16 

14 

5G 

Ans,  The  boys  24,  4,  and  12  days,  respectively,  and  the 
man  16. 


EYOLUTION. 


SQUARE   ROOT. 

(434,  page  319.) 
Ex.  9.     Ans.  234135. 

(441,  page  321.) 
Ex.  3.    200  X  1|=225  sq.  yd. ;  V225=15  yd.=45  ft,  Ans. 


Ex.4.     10  A.=1600  rd. ;  i/l600=40  rd.,  length  of  ono 
side  ;    40  x  4  =  160  rd.,  Ans. 

Ex.  5.     45^=2025,  square  of  the  base  ; 

60'=3600,  square  of  the  perpendicular; 

5625,  square  of  the  hypotenuse. 
V'5625  =  75,  Ans. 


SQUARE   ROOT.  135 

Ex.  6.     39^—1521,  square  of  the  hypotenuse 
15'=  225,  square  of  the  base  ; 

1296,  square  of  the  perpendicular. 
V'l296=:36,  height  of  the  stump. 
36  ft.  +  39ft.=:'75  ft.,  Ans. 


Er.  1      |/40'— 33'=22.60  +  ,fronafootofla(idertocjncsidei 
4/40'  — 21'=34.04-f      "      "     "      "     "    other" 
56.64  +  ,  ^^5. 

Kx.  8      5 2' =2 704,  square  of  hypotenuse  ; 

48^=2304,  square  of  perpendicular ; 

400,  square  of  the  base ; 
|/40b=20,  Ans. 


Ex.  9      1  mi. =320  rd.,  length  of  1  side  of  the  park. 
820''=102400 
102400 


204800;    |/204800=452.5-f ,  diagonal. 
320  X  2  =  640,  distance  around  the  park  to  the  opposite  corner. 
640— 452.5  =  18*7.5,  distance  between  A  and  B,  when  A  ar- 
rives at  the  opposite  corner.     187.5  4-^2  =  93  7+,  Ans. 

Ex.10.  20Hl6'=square  of  the  diagonal  of  the  floor.  The 
diagonal  of  the  floor  and  the  height  of  the  room  will  form  the 
base  and  perpendicular  of  a  right-angled  triangle,  of  which  the 
diagonal  from  the  lower  corner  to  the  opposite  upper  corner 
IS  the  hypotenuse.     Ilence 

ini     c^^^  r  square  of  the  diasjonal  of  the  floor  ; 
lb  =iioo  j  . 

1 2*= 144  square  of  the  perpendicular ; 

800  square  of  the  required  diagonal ; 
i''80b  =  28.28427l+  feet,  Ans, 


186  EVOLUTION. 

Ex.  11.     2:3: :  (63.39)'  :  (     )=:6027.43815 
1/6027.43815  =  77.63  -|-rods,  Ans. 

Ex.12      1:2::5':(     )  =  50 ; 

4/50  =  7.07106 +feet,  Am. 

CUBE    ROOT. 

(444,  page  327.) 

APPLICATIONS  IN  CUBE  ROOT. 


Ex.  1.     Vl331=:ll  ^t,,Ans. 


Ex.  2.    V373248=:72  iii.=6  ft.,  Ans. 


Ex,  3.     V474552  =  78  in.=6i  ft.,  length  of  1  side ; 
61x61  =  421  sq.  ft.,  Ans. 

Ex.  4.  V||=f  ft.,  length  of  1  side  ; 
J  X  I  sq.  ft. = area  of  1  side  ; 
fxjx6xi=f  sq.  yd.,  area  of  6  sides,  Ans. 

Ex.  5.  If  the  bin  be  divided  by  a  vertical  section  equi- 
distant from  the  ends,  the  two  parts  will  be  cubes,  each  of  a 
capacity  of  one  half  of  the  bin. 

125  X  2150.4  =  268800  cu.  in.  in  the  bin ; 
268800-^2  =  134400  cu.  in.,  contents  of  one  halfl 
Vl34400=51.223+in.,  width  and  depth. 
61.223  X  2  =  102.446  in.,  length. 

Ex.7.     1  :2  =  (14.9)»:  (     )  =  6615.898 
^''¥6157898= 18.7  + inches,  Ans, 

Ex.  8.     16  :  25  • :  8  :  (    )=12.5  cube  of  the  diameter. 

Vl2^  =  2.32+ ft. 


ARITHMETICAL   PROGRESSION.  137 


ARITHMETICAL  PEOGRESSION. 

(451,  page  329.) 
Ex.1.     (19~l)x3=54;  54  +  4=58,  ^W5. 
Ex.  2.     (13  —  1)  X  5  =  60  ;  75  —  60  =  15,  Ans. 

Ex  3.     2=firstterm;  3  =  com.  diff. ;  18=No.  terms. 
(18  — l)x3  =  51;  514-2  =  53  cents,  ^/i5. 

Ex.4.     (40-l)xi=9f;  9} -hi  =  101,  Ans. 

Ex.  5.     20= first  terra  ;  5= com.  diflf. ;  9= No,  terms. 
(9  — l)x5  =  40;  40  +  20  =  60,  ^W5. 

Ex.  6.     100=first  term  ;  7=com.  difF. ;  46=No.  terms. 
(46  — l)x  7  =  315;  315  +  100=$415,  ^W5. 

(452,  page   330.) 

Ex.  1.     17-2  =  15  ;  15-^5  =  3,  Ans. 
Ex.  2.     14—2  =  12  ;  12-~6  =  2  years,  Ans. 
Ex.  3.     501-1  =  49^  ;  49i-v-33  =  li,  Ans. 
Ex.  4.     3=first  term  ;  9J=last  term  ;  14 =No.  terms. 
91-3=61;  61-^-13  =  1  com.  dif. 

(453.) 

Ex.1.     43—7—36;  36-^4=9;  9i-l  =  10,  Ans. 
Ex.  2.     40-21  =  371 ;  37iv-7i=5  ;  5  +  1=6,  Ans. 

Ex.  3.     6=first  term  ;  226  =  last  term  ;  4= com.  diff. 
226  —  6  =  220;  220-^4  =  55  ;  55  +  1=56,  ^»#. 

(454,  page  331.) 
Ex.  1.     (5+32)xV=222,  Ans. 
Ex.2.     (1+12)  x\^=1 8,  Ans. 


18b  GEOMETRICAL    FfioQRESSION. 

Ex.  3.     24=:first  term;  1224=:last  term;  52=:No.  terms. 
($1224  +  124)  X  ^/=$32448,  Ans. 

Ex.  4,     4= first  term,  or  twice  the  distance  to  the  first  apple, 
400  =  last     "       "       "      "         "  "         last     " 

100= No.  terms. 
(400  yd.  +  4  yd.)  x  4^^^  =  20200  yd.,  Ans. 


GEOMETRICAL  PROGRESSION. 
(458,  page  333.) 
Ex.  1.     4x3'=26244,  Ans. 
Ex.  2.     1024  X  (iy=jV3%\=T\,  ^ns. 
Ex.3.     1=  first  term;  2= ratio;  9= No.  terms. 
1  mill  +  2''=256  mills=$.256,  Ans. 

Ex.  4.      1  X  U)    =77J  =  ;n=r-7T7n;j  ^^s. 

Rx.  5.     1=  first  term;  1.07 = ratio  ;  5= No.  terms. 
1  X  (1.07)*=|1.40255+,  Ans. 

Ex.  Q.     3= first  term;  3= ratio;  7= No.  terms. 
$3x3"=$2187,  Ans. 

(45 9,  page   334.) 

Ex.  1.     (512x3)--2  =  1534;  1534—2  =  767,  Am. 
Ex  2      (262144  x4)-4  =  1048572; 
1048572-^3  =  349524,  Ans. 
Ex.  3.     (162  X  3)--2=484  ;  484-^2=242,  Ans. 

Ex.  4.    J^-^^=l,  ratio; 

(1x5)— 0  =  1;  l^4=i,  Ans. 


PROMISCUOUS   EXAMPLES.  139 

Ex.  5.     2— first  term;  6-^2  =  3,  ratio  ;  12=^0.  terms. 
$2  X  3"  =  $354294,  last  term  ; 
($354294  X  3)  — 2r=$1062880  ; 
$1062880-v-2=$531440,  Ans. 

Ex.  6.     7:=  first  term,  or  yield  of  the  first  year; 

7= ratio ; 

12 ^No.  terms,  or  the  number  of  years  to  yield. 

7x'7"  =  '7^'=:13841287201,   last   term,    or    12tb 

year's  produce. 

(13841287201  x7)  — 7  ,   „ 

^^ ^ =16148168400,  sum  of  all  terms. 

0 

16148168400-4-1000  =  16148168.4  pt. 
16148168.4  pt.  =  252315  bu.  41  qt.,  Ans, 

Ex.  7.      200-^20  =  10,   the   number   of  times  the  family 
doubled  its  number. 

10+1  =  11,  No.  terms;  2  ratio. 
6x2^'»=6144,  ^/i5. 


PROMISCUOUS  EXAMPLES 


(Page  334.) 

Ex.  1.     800  x  2  =  1600,  the  sum ;  and 
200  x  2=  400,  the  diflference. 
The  greater  of  any  two  numbers  is  equal  to  the  less  -f-  ths 
difference ;  and  the  greater  and  the  less,  or  the  sum   of  the 
numbers,  must  be  composed  of  twice  the  less  and  the  differ- 
ence.    Hence 

1600  -400  =  1200,  twice  the  less; 
1200-^     2=  600,  the  less  ; 
6004-400=1000,  the  greater. 


140  PROMISCUOUS   EXAMPLES. 

Ex.  2.     I  of  j\=fy.     If  ^<L.  of  a  number  be  added  to  itsell, 
the  result  must  be  l/j  times  the  number.    Hence, 
61-M/y  =  55,  Ans. 

Ex.3.  3  h.  21  min.  15  sec.=:12075  sec;  1  da. =86400  sec; 

ifnMa.=TTV2  dx,Ans, 
Ex.  4.       3  bu.  3  pk.  3  qt. 
10 


38  bu.  1  pk.  6  qt. 

1 


269  bu.  0  pk.  2  qt.,  Ans. 

Ex.  5.     A  and  B  together  have  3  times  A*s ; 

C  and  D  together  have  $300  +  $500=$800  ; 
And  they  all  have  3  times  A's+$800. 
Therefore,  $1100— $800=1300=3  times  A's. 
$300-^3  =  $100,  Ans. 

Ex.  6.     B  has  A's  votes  H-  20O 
C  has  A's  votes +  1000 
-  A  B  and  C  have  3  times  A's  votes +  1200 
Therefore,  3000  —  1200  =  1800,  3  times  A's  voteii. 
1800-^3  =  600,  Ans. 

Ex.  7.     —-7=:—-,  Ans. 
I7i     70' 

Ex.  8.  J— ^  =  j'g.  Hence  10  is  j\  of  the  number;  and 
the  number  must  be  560,  Ans. 

Ex.  9.     $28.35-r-$.35  =  81  gal.  mixture. 

81  —  63  =18  gal.  water  added. 

Ex.  10.  When  A  had  gained  },  he  had  f  of  the  original 
stock.  B,  after  his  loss,  had  ^  as  much,  or  f  of  the  original 
etock ;  hence  he  had  lost  | ;  the  $200  which  he  lost  was  | 
of  his  stock;  and  his  whole  stock  must  have  been  200-r-}  = 
8500. 


PROMISCUOUS   EXAMPLES.  141 

Ex.  11.    1.35  X  13  ==$4.55;  $31.55  — $4.55=:$27,  cost  of  the 
whole,  if  the  wheat  had  been  at  the  same  price  as  the  barley. 
17  +  13  =  30,  whole  number  of  bushels. 
$27-^30  =  1  .90,  price  of  barley,  )    , 


':\- 


$.90 +  $.35  =$1.25,  price  of  wheat, 

Ex.  12.  4  mo.  11  da.    7  h.    5  min. 
3  20         15       21 

21  da.  15  h.  44  min.,  Ans. 
Note. — Borrow  31  days  for  March. 

Ex.  13.  The  point  of  time  divides  the  whole  12  hours  into 
two  intervals,  which  are  in  the  ratio  of  9  to  11.  Hence,  by 
Partnership, 

9 
11 

20  :  9  : :  12  h.  :  (     )  =  5  h.  24  min.,  Ans. 

Ex.  14.  The  least  common  multiple  of  63,  42  and  31| ;  or, 
since  63  is  2  times  31^,  the  least  common  multiple  of  63  and 
42,  which  is  126,  Ans, 

Ex.  15.  The  least  common  multiple  of  8,  9,  15  and  16, 
which  is  720,  Ans. 

Ex.  16.  Since  B  gets  in  debt  $10  yearly,  his  income  would 
enable  bin:   to  spend  $30  — $10  =  $20  a  year  more  than  A 
8})ends.     Hence  $20  is  }  of  the  income  ;  and 
$20  X  8=:$160,  income,  Ans. 

Ex.  17.  $2.19  xf-f If  =  $2.40,  Ans, 

Ex.  18.     I  \:       ]\  \',  $3.37|  :  (     )=$52.779,  Ans, 
I5  )        I2  ) 

Ex.  19.  $1000  :  $200  :  :  6  mo.  :  (     )  =  li  mo..  Am, 
K.  P.  7 


142 


PROMISCUOUS    EXAMPLES. 


Ex.20.  |2350.80-^.40  =  $5892,  left; 

$5892 +  $2356.80=18248.80,  Ans. 

Ex.  21.  — ^^=tVo  X  f  X  f =1  =  121  per  cent.,  Ans. 


Ex.  22.  1  private  has  1  share  ;  60  privates  have  60  shares  . 

1  subaltern"  2       "  6  subalterns "    12       '* 

1  lieut.        "  6       "  3  lieut's         "18      " 
1  commander  has     12      " 


All  have  102  shares, 
$1 0200 -f- 102  =  $  100,  share  of  a  private. 
$100  X  12       =$1200,  share  of  the  commander. 

Ex.23.  19  —  16  =  3;  51-^3  =  17  hours,  ^?Z6\ 


Ex.  24.  40  < 


20 

2V 

14 

14 

7 

133^ 

30 

tV 

10 

10 

5 

95 

50 

tV 

10 

10 

5 

95 

54 

tV 

20 

20 

10 

190 

Ans, 


Ex.  25.  $33.75-v-22.5=$1.50,  selling  price  per  bu. 
$22.50-^18       =$1.25,  buying    "  " 

$  .25,  profit  on  1         " 
$.25x240=$60,  Ans, 


Ex.  26.  Tlie  wagon  is  worth  4  times  the  harness  ;  the  hursu 
is  worth  8,  times  the  harness;  hence  the  horse,  wagon  and 
harness  together  are  worth  84-4  +  1  =  13  times  the  harness 
Tnerefore,  $169-^13  =  $13,  harness,  ^?Z5. 

Ex.  27.  18  in  :  42  ft.  :  :  40  in.  :  (     )  =  93i  ft.,  Ans. 

Ex.  28.  25  rd.  :  40  rd.  : :  4  rd.  :  (     )  =  6|  rd.,  Ans. 


PROMISCUOUS  j:xamples.  143 

Ex.  29.  J,  /j,  and  I2=|f,  if,  and  if 
And  since  fractions  having  a  common  denominator  are  pro- 
portional to  their  numerators,  we  have 

1 5  shares  for  A  and  B ; 

18      "       "    A  "    C; 

13      "       "    B   "    C; 

46,  twice  the  number  of  shares  for  A,  B  and  C. 
4b  -,'2   =23  shares  for  A,  B,  and  C. 
23-13  =  10      "       "    A; 
23-18=  5      "       "    B; 
23-15=   8      "       "    C. 


$26.45  X  if  =  $11.50,  A.'s  portion  ; 
$26.45  x^^-^zS  5.Y5,  B's        "        j 
$26.45  x/^=$  9.20,  C's        " 


Ans, 


Ex.  30. 


6^  [  •  ^^4  [   •  •  12  :  (     )=480  oz.  =  30  lb.  Ans. 


Ex.  31.  $6300  x  I  =  $  900,  A's, 

$6300xi=$1260,  B's; 

$6300x|  =  $1400,  Cs; 

$900  +  $1400       =$2300,  D's  ; 


$6300— $5860=$440 

$440  xf=$l 65,  E's, 
$440xf=  275,  Fs. 

Ex.  32.     $200xl.593848=$318.769  +  ,^w*. 

Ei,  33.  At  the  time  of  the  dismissal,  the  provisions  on 
hand  would  supply  360  men  1  month  ;  they  would  supply  ^ 
us  many  men  5  months. 

Hence  360-4-5  =  72,  the  number  that  remained; 
360-72  =  288,  dismissed,  Ans. 


144 


PROMISCUOUS   EXAMPLES. 


Ex.  3i.     $1.338220 ^amt.  of  $1  at  compound  int.  for  6  yra. 
at  6  per  cent. 
$669,113-^1.338226  =$500,  principal. 
$669.113— $500  =  $169.1 13,  interest. 
$500  X. 06  =  $30,  simple  int.  of  $500,  for  1  year 
at  6  per  cent. 
169.113-^30=5.6371  yr.  =  5  yr.  7  mo.  19.356  +  da.,  Am, 

Ex.  35.     $148.352-f-9.728=$15.25,  Ans, 

V 

Ex.  36.  It  is  evident  that  the  product  of  two  numbers  must 
»'ontain  each  common  factor  to  the  two  numbers  twice,  and 
'^ach  factor  not  common  once. 

483-f-23  =  21,  product  of  the  factors  not  common. 

23x23x21  =  11109,  Ans. 

Ex.  37. 


^} 

12) 

2 

n 

:       9 

::  1  :2 

12 

8 

15 

(  ) 

9 
(    ) 

1 
15 

9 

140 

Ex.  38. 


Ex.  39. 


15f,  Ans. 
36  )    .  60  )        36  )       (     )  ) 
9r   •27[--     1^-      11  f 
Ans,  144  yards. 

1    X  4  =  4,  A's  product ; 
3    x2  =  6,  B's         " 
7ixl  =  7|,  C's       " 

A  '  2 

171  :  4  : :  $52.50  :  (  )=$12,  A's  share ; 
17^  :  6  : :  $52.50  :  (  )  =  $18,  B's  share; 
I7i  :  7i  : :  $52.50  :  (    )=$22.50,  C's  sham 

Ex.40.     |andf  =  fandf. 
4  +  5  =  9. 
9  :  4  ::  $45  :  (     )=$20,  A's; 
9:5::  $45  :  (    )=$25,  B's. 


PROMISCUOUS    EXAMPLES.  H5 

Ex.  41.     $.35,  interest  of  |1  for  5  years  at  7  per  cent.; 
$33.25-^. 35=$95,  Ans. 

Ex.  42.     6  X  VS  =  6  X  'VY^O  X  1  =  3,  Ans. 
Ex.43.  2  +  3+4  =  9. 


9:2::  $360 
9:3::  $360 
9:4::  $360 


(     )  =  $  80 

(     )i=$120  [•  Ans. 

{     )=r$160, 


Ex.44.     8i:6f  ::12i  :(    )=9^^,  Ans. 

Ex.45.  9  X  9=::81  sq.  in.  in  1  stone  ; 

144  X  9  =  1296  sq.  in.  in  1  sq.  yd. ; 
1296-^81  =  16  stones  for  1  sq.  yd. ; 
16x40  =  640,  Ans. 

Ex.46.  1.00— .08=3.92;  $2  3 -+.92==  $25,  cost  of  the  calves 
and  sheep  sold.  Since  the  price  multiplied  by  the  quantity 
gives  the  cost,  we  have  these  two  conditions,  viz. ;  3  times  the 
number  of  calves,  plus  2  times  the  number  of  sheep,  equals 
76  ;  and  3  x  J  =  J  of  the  number  of  calves,  plus  2  x  f  =  f  of 
the  number  of  sheep,  equals  25.  Expressing  these  conditional 
as  in  Analysis,  page  294,  we  have 

C.  8. 

1st  condition  3         2         76 

2d  condition  f         I         25 

2dx4=  3       Y       1^^ 

Subtract  the  1st  from  the  3d=  |         24 

That  is,  I  of  the  sheep  are  equal  to  24  ;  hence 

24-7-|  =  20,  number  of  sheep. 
76  — (2  X  20)^:36  ;  36-7-3  =  12,  number  of  calve*. 

Ex.  47.     Hi  I      16 

lOi  J-   :    7  ^  : :  546  :  (    )  =  384,  Am. 


13    )       15 


146  moMiscuous  examples. 

Ex.  48.      12     )       (     )  ) 

15f  V  :     15  V  ::  2  :  1. 
9    1  1) 

Reducing  the  statement,  (     )  =  8,  the  number  of  men  re* 
quired  to  finish  the  job. 

12  —  8  =  4  men  withdrawn. 

Ex.  49.  It  is  evident  that  to  increase  the  number  in  both 
rank  and  file  by  1  man,  would  require  twice  the  number  in 
rank  or  file  at  first,  plus  1  (for  the  man  at  the  corner).  And 
since  to  eftect  this,  required  59  +  84  =  143  men,  ^^^^=^=^1  is 
the  number  of  men  in  rank  or  file  at  first.  Hence 
IV -\- 59  =  5100  men  under  command,  Ans» 

Ex.  50.  Cost  of  the  corn  was  2  times  the  cost  of  the  barley 
"       "       wheat  "    4     "        "       "      "  " 

Cost  of  corn  and  wheat  "    6     "        "       "      "  " 

Hence,  cost  of  barley  was  }  of  the  cost  of  com  and  wheat 
Again, 

Wheat  cost  $243  -f  J  of  the  whole. 
Corn       "       153 +  yV     "         " 
Wheat  and  corn     "     $396+1       «         « 

Barley    "         QQ-^t\     "        "     ( J  of  w.  iSz;  c.) 

Cost  of  the  whole  was  $462  +  j\     "         " 
Hence,  $462  was  |f— y\r=|i  of  the  whole  cost  • 
And       $462-+||=$'756,  the  whole  cost. 
And  since  the  barley,  corn,  and  wheat  cost  in  the  proper 
lion  of  1,  2,  and  4,  we  have 

1+2+4=7 
$756x4  =$108,  cost  of  barley;  $108-+$  .60  =  180  bu.  barley; 
$756  X  ^  =  $216,  cost  of  corn;    $216^$  .75  =  288  bu.  corn  ; 
1*756  X 4  =$432,  costof  wheat;  $432-+$1.50  =  288  bu.  wheat; 

Ans,  756  bu.  grain. 


6 

a 

u 

and  7 

(i 

u 

21 

$630  X  2V 

=  $240, 

1st; 

$630  X/y 

=$180, 

2d; 

$630  X  2V 

=$210,  3d. 

PROMISCUOUS    EXAMPLES.  147 

Ex  51       As  often  as  the  first      has  1, 

"     second   "  f, 

and     "     third      "    iofl5=|; 
1,  f,  and  f =1,  f,  and  f. 
We  therefore  assume  8  as  the  proportional  number  for  the  first ; 

"  •      "     second; 
"  "     third 


Ans, 


Ex.52.     $28xl.20=$33.60  what  the  56   remaining  gal- 
lons must  be  sold  for.  $33.60-^-56=$.60,  Ans. 

r.     .0        rVoXf         20      4     1     4     2      16       ^^.     . 
^^''''    ^^^irT^0><5^2><3><3-225=-^^ 

Ex.  54.  $3500— $2100=$1400,  what  B  owns  now  ; 
$1400^  1.40=$1000,  B  put  in  ; 
$2100-T-     1.40  =  $1500,  C  put  in. 

Ex.  65.     i/S'-f  16^  =  17.88 +ft.,  Ans. 
Ex.  56.     12  )       10  )       ^.  ^       .     V  J 

Ex.  57.     $12-v-1.09=$11.0091-f,  worth  of  sugar; 
$12— $11.0091=$.9909  +  ,  grocer's  profit; 
$6^1.10=$5.4545+,  worth  of  beef; 

$6— $5.4545=$.5455+,  farmer's  profit; 
$.9909— $.5455 =$.445+,  grocer  gains  more, 

Ex.  58.     $336.42  — $311.50  =  $24.92,  interest; 

$4.15i  int.  of  $311.50  for  1  yr.  4  mo.  at  1  per  cent 
$24.92-i-$4.15|^=6  per  cent,  Ans. 


148  PROMISCUOUS    EXAMPLES. 

Ex   50       A    2    «,-,/]  2 — 12    10    anri  -8- 

r^A.   OJ.         513?    '*"'^    5  —  15)    1  5>    ^l^'-i    1  5» 

These  fractions  are  to  each  other  as  their  nuiaerators,  12,  10, 
and  6 ;  and  these  numerators  are  to  each  other  as  6,  5,  and  3, 
Hence,  we  have  6  shares  for  A  and  B ; 
5  shares  for  A  and  G  ; 
0  3  shares  for  B  and  C. 

14,  twice  the  number  of  shares  for  A,  B,  an  J  C. 
14 -7- 2  ==7  shares  for  A,  B,  and  C. 
7  —  3  =  4  shares  for  A ; 
7  —  5  =  2  shares  for  B ; 
7  —  6  =  1  share  for  C. 
Hence  $20  x  -f =$llf ,  A's  ; 

$20xf  =  $54,B's; 
$20x|=$24,  C's. 

Ex. .60.     $375-^.025  =  815000,  Ans. 

Ex.  61.     Interest  commenced  Apr.  1,  1857. 

Amt.  of  note  July  1,  1857,  (3  mo.) $1015 

Payment, 560 

New  Principal, $  455 

Amt.,  Dec.  1,  1857,  (5  mo.) 466.37  -f 

Payment, 406 

New  Principal, $     60.37  4- 

Amt.,  Aug.  23,  1859,  (1  yr.  8  mo.  22  da.).  .$     66.63+, 

Ans. 
Ex.  62.  B  has  |  x  ^  of  A's  =f    of  A's  ; 

C  has  4  X  f  of  B's=-^  x  f  x  f  of  A's  =|f  of  A's  ; 
D  has  I  X  f  of  C's= J  x  f  x  |?  of  A's=-;  f  of  A's 
And  since  D  has  $45  more  than  C, 

If  of  A's— 1^  of  A's=$45;  or  ^%  of  xVs  -$45. 
Hencc;  $45-^/^   =$378,  A's;] 

$378  X  I   =$336,  B's; 
$378x|^=$360,  C's;   ^^^^• 
$378  X  If  =$405,  D's. 


PROMISCUOUS  EXAMPLES. 


149 


Ex.  63.  B  had  the  use  of  $300  for  2Y  months  before  it  waa 
dne,  which  was  equivalent  to  the  use  of  $1  for  27  x  800=8100 
months.  But  the  use  of  $1  for  8100  months  is  equal  to  the 
use  of  $600  for  VoV-=13i  months,  the  time  he  should  wait. 

Ex.  64.  His  savings  and  expenses  together,  or  his  sal&ry 
is  If  +  A  =  Tf  ^^  what  he  saves  ;  hence  $800  =  if  rf  That  ^ie 
aves;  and  $800-^|f =$550,  ^/i5. 

Ex.  65.     $.87i=$J  ;  $1.00 H- 1.1 0=:$|^,  cost  per  yd.; 
$-;^— $1  — $JL.,  loss  at  $.871  per  yd. ; 
Tj^TT^/o^.OSf  =  3^  per  cent.,  ^;25. 


Ex.  66. 


9 1  JL         2  4  0 
^^ 1  I  11. 


240 
'320 


3       /3\*     27      , 


^     ^^        63       189     27        /27     3     , 
^"'^-     1491=448=64'    / 64=?  ^"^• 


Ex.  68.     50  - 


20 

tV 

20 

2 

2 

20 

35 

tV 

10 

2 

2 

20 

60 

To^ 

15 

3 

3 

30 

70 

^V 

30 

3 

3 

"lo" 

30 

100 

Ans,  20  of  oats  and  corn,  and  30  of  rye  and  wheat. 


Ex.  69. 


40  +  500 


=  270,  half  the  sum  of  the  extremes. 


And  since  the  sum  of  all  the  terms  is  equal  to  half  the  sum  of  the 
extremes  multiplied  by  the  number  of  terms,  6480-7-270=24, 
the  number  of  creditors. 

500  —  40=460,  difference  of  extremes. 
$4604-23 =$20,  common  difference. 


Ex.  70.  Vl28''-f  72'  =  146.86-h,  Ans. 


150 


PROMISCUOUS   EXAMPLES. 


10 
11 

14 

4 

10 


550,  Ans. 


Ex.  71.  If  7  pounds  of  butter  are  equal 
to  10  pouuds  of  cheese,  11  pounds  of  but- 
ter are  equal  to  y  of  10  pounds  of  cheese ; 
and  if  2  bushels  of  corn  are  equal  to  11 
pounds  of  butter,  14  bushels  of  corn  are 
eqial  to  'g*-  of  11  pounds  of  butter,  or  y- 
of  y  of  10  pounds  of  cheese  ;  and  if  8  bushels  of  rye  are  equal 
to  14  bushels  of  corn,  4  bushels  of  rye  arc  equal  to  f  of  14 
bushels  of  corn,  or  |  of  Jg^  of  Jy  of  10  pounds  of  cheese  ;  and, 
finally,  if  1  cord  of  wood  is  equal  to  4  bushels  of  rye,  10  cords 
of  wood  are  equal  to  10  times  4  bushels  of  rye,  or  \^  of  f  of 
V  ^^  V  ^^  ^^  pounds  of  cheese=550  pounds  of  cheese,  Ans, 

Note. — ^Instead  of  the  fractional  form  the  vertical  lino  may  bo  iSQd^ 
as  above. 


Ex.  72.  $18-^f  =  $45        A's  gain. 

f  :  ^  ::  $45  :  (     ) =$37.50,  B's  gain. 

$45-^.06      =$750,  A's  stock  ; 
$37.50-^.06  =  $625,  B's  stock 


'.  73.  20  )       (     )  )         30 ' 

45^ 

21  V  :     25  V  ::  15 

16 

10)           8)        12 

^-  18 

>■ 

3^ 

5 

Reducing  the  statement,  we  have 

(     )-=84 

,  Ans, 

Ex.  74.  2|  :  27^  ::  10  ft.  :  (     )  =  103i  it.,  Ans. 


Ex.  75.  A  can  do  ^  of  the  work  in  1  day  ; 

J^  u        ^      a  u        a    -j^     u 

C       "    ^^    "         "      "   1    " 
They  all     "      i  + 1  +  7*2  =  f  of  the  piece  in  1  day. 
Hence  it  will  require  1-^f =f  of  a  day. 


PROMISCUOUS   EXAMPLES.  151 

Ex.  Y6.  $1890-^1.25=:$1512,  true  value  of  1st. 
$1890—   .75=$2520,    "         "      "2d. 

$4032,    «         "      "  both 
$1890  X  2  =$3780,  received  for  both. 

Ans,  $  252,  lost. 

Ex.  77.  If  C  paid  one  half  the  cost,  A  and  B  together  paid 
$50.  Since  C  cows  eat  as  much  as  4  horses,  12  cows  eat  as 
much  as  8  horses.  Therefore,  A  put  in  9  horses  for  1  unit  of 
time,  and  B  put  in  the  same  as  8  horses  for  2  units  of  time. 
Hence, 

A's  use  of  the  pasture  was  9x1=  9  horses  for  1  unit  of  time  ; 
B's    "       "  "         "    8x2  =  16      "    •«    1     "     «     " 

A  andB's  use  of  the  pasture  was =25      "       "    1     "     "     " 

$50x^5  =$18,  A  paid  ; 

$50  X  if  =$32,  B  paid. 
Again,  C's  time  was  2i  times  B's  time,  or  2  x  2^  =  5  units  of 
time.  And  since  C  paid  half  the  cost,  his  use  of  the  pasture 
must  have  been  as  much  as  A's  and  B's  together,  which  is  25 
horses  for  1  unit  of  time  ;  and  this  is  equal  to  5  horses  for  C's 
5  units  of  time,  Now  the  pasturage  of  1  horse  is  f  times  the 
pasturage  of  1  cow,  and  the  pasturage  of  1  cow  is  J/-  times  the 
pasturage  of  one  sheep ;  hence  the  pasturage  of  5  horses  (for 
which  C  paid)  is  '/xf  x5=25  times  the  pasturage  of  1 
sheep.     Therefore  C  put  in  25  sheep. 

Ex.  78 
$350C-4-1.0175=$3439.803-f ,  pres.  worth  of  1st  installment; 
13500 -M.02i   =$3420.195+,         "  "  2d  ** 

|3500-^l.04|   =$3343.949+,  "  "  3d  ** 

$10203.947 -f,  Ans. 

Ex.  79.  Had  the  farmer  sold  both  geese  and  turkeys  at  $.75 
%piece,thc50  fowls  would  have  brought  $.75  x  50 =$37.50, 


152  PROMISCUOUS   EXAMPLES. 

which  is  $52.50 — $87.50  =  $15  less  than  they  really  brought ; 
consequently  the  difference  between  the  two  estimates  of  the 
turkeys,  reckoned  at  $.75  and  $1.25  apiece,  is  $15.  Hence. 
$].25-$.75  =  S.50;  $15-^$.50=:30,  the  number  of  turkeys; 
and  50  — 30=::20,  the  number  of  geese. 

Ex.  80.  B  gains  of  A  3  miles  an  hour,  and  C  5  mih:a 
an  hour.  Hence  B  will  pass  A  every  ^  hours=24  h.  ?0 
inin.=:1460  min. ;  and  C  will  pass  A  every  y  hours=14  h, 
36  min. rr: 876  min.  Now  the  least  common  multiple  of  1460 
and  876  will  express  the  number  of  minutes  in  which  B  and 
C  will  first  pass  A  together. 


2,2 

1460  . 

.  876 

73 

865  . 

.  219 

5,3 

5  . 

.  3 

2x2x73x5x  3  =  4380  min.=:6  da.  1  h.,  Ans. 

Ex.  81.  i,  -}  and  i^f  f,  ii  and  i|. 
And  since  fractions  having  a  common  denominator  are  to  eacV 
other  as  their  numerators.  A,  B  and  C  were  to  share  in  the 
proportion  of  20,  15  and  12.  But  C  dying,  his  12  parts  must 
be  shared  by  A  and  B  in  the  proportion  of  20  and  IS,  or  4 
and  3.  4  +  3  =  7. 

7  :  4  : :  12  :  (     )  =  6f ,  A's  share  of  C's  12  parts ; 

7  :  3  ::  12  :  (     )  =  5i,  B's     "      "    "     "     " 

20  +  6-|==26|,  A's  number  of  parts  of  the  money ; 
15  +  51  =  20 j,  B's       "         "      "         "  " 


47 

:  26 

f  :: 

$100000  : 

( 

)= 

:$57l42.85f  A's,  ) 
.$42857.14f,  B's,  )         • 

47 

:  20 

\' 

:  $100000  : 

( 

> 

Ex. 

82. 

A' 

C 

c 

s  +  B's=:5 

5-fB's=:7 

s-B's=l 

Since  A's  +  B's  are  t< 
B's  +  C's  as  5  to  7,  A 
and  B  together  have  5 

i.±J-  =.4,  C's 

proportion ;    as  often  as  B  and  C  to- 

7- 

-4r=3,  B's 

u 

gether  have   7.      And 

5- 

-3  =  2,  A's 

u 

since  C's -B's  are  to  C's 

PROMISCUOUS   EXAMPLES. 


158 


f  B's  as  1  to  7,  7  is  the  sum  and  1  the  difference  of  B  and  C's 
proportionate  shares.  Hence  we  find  the  proportionate  share 
of  each.     Then 

2  +  3  +  4  =  9,  the  sum  of  their  proportions. 

And  A  has  |  of  135  =  30  sheep ; 
B     "   f  "  135=45      « 
C     "    ^  "  135  =  60      " 


Ex.  83.  250x4  =1000 
300x4^  =  1350 
369x5   =1845 


919 


4195 


4195-4-919  =  4f  II,  Ans. 


Ex.  84.  The  relative  values  of  the  work  performed  by  one 
of  each  class,  in  the  same  number  of  days,  are  as  follows  : 
1  boy        3x    8  =  24  )        (2 
1  woman  4  x    9  =  36  >■  or  •]  3 
1  man       6  x  12  =  72  )        (  6 
The  relative  amounts  of  wages  received  by  the  whole  num- 
ber of  boys,  women  and  men,  in  the  same  number  of  days,  are 
as  follows : 

Boys,  $5  ;  women,  $10  ;  men,  $24. 
Hence  the  proportions  of  boys,  women  and   caen  ar*^  ex- 
pressed by  the  following  quotients : 

5-^2  =  21  boys; 
10-^3=3i  women ; 


24- 

i-6=4    men. 

n 

:2i  :: 

59  :  (     )  =  15  boys; 

H 

:  31  : 

59  :  (     )  =  20  women; 

n 

;  4    : 

59  :  (     )  =  24  men. 

7* 


164  PROMISCUOUS   EXAMPLEa 

Ex.  85.  A,  B  and  C  fill  ^   of  it  in  1  hour ; 

B,  C    "    D   "    1      "     "    " 

C,  D    "    A  "  tV     "     "     " 

D,  A   "    B   "  j\     "     "    « 


A,  B,  C  and  D  fill  i  of  J/^=:-JJ^  of  it  in  1  hour. 
W,  X  and  Y  empty  |  of  it  in  1  hour ; 
X,  Y    "    Z      «      }     "     "    ** 
Y,  Z      ''    W    "      \     "     "     " 
Z,  W    "    X     "      1     "     "    " 

W,  X,  Y  and  Z  empty  J  of  111= Jt?^  of  it  in  1  hour. 

yVo"~TVo  =tVo>  ^^  emptying  pipes  gain  of  the  filling  pipes 
m  1  hour.  Therefore,  to  exhaust  the  fountain  will  require 
120-M9  =  63V  hours,  Ans, 

Ex.  86.  1  A.  1  R.  6  P.  181  gq.  yd.=56250  sq.  ft.; 
56250        ^      , 


Ex.  87. 


75x125   ' 

U^lbO, 

$   Ix 

0  =  $    0 

2x 

1  = 

2 

4x 

2  = 

8 

8x 

3  = 

24 

16  X 

4= 

64 

32  X 

5  = 

160 

64  X 

6  = 

384 

128  X 

7  = 

896 

256  X 

8  = 

2048 

512X 

9  = 

4608 

1024  X 

10  = 

10240 

2048  X 

11  = 

22528 

Cost,  $4095.  $40962,  sum  of  products. 

40962 -f-4095  =  10  mo.,  average  term  of  credit. 
Jan.  1  4-10  mo. = Nov.  1,  average  time. 


PROMISCUOUS   EXAMPLES.  155 

Ex.  88.  44.32  x  36  =  1595.52  sq.  ch.  =  159  A.  2  R.  8.32  P.. 

Ans, 

Ex.89  l-|-i4-3-=lf.  That  is,  if  a  number  be  increased 
by  ^  and  ^  of  itself,  the  result  will  be  If  times  the  number. 
Hence,  by  the  conditions.  If  times  the  number,  plus  18,  is 
equal  to  2  times  the  number;  consequently,  18  is  2  — l|  =  i 
of  the  number,  and  18  x  6  =  108  is  the  number. 

Ex.  90.  If  that  which  is  worth  $.621  be  rated  at  $.56,  what 
ought  that  which  is  worth  $.25  be  rated  at  ? 

$.621  :  $.25  ::  $.56  :  (    )=$.224 
Therefore,  m  the  barter  for  a  pound  of  coffee  at  $.22,  the  mer- 
chant obtains  that  which  is  worth,  ratably,  $.224;  hence  he 
gains  $.004  on  $.22  ;  and  $.004-f-$.22  =  .01/y,  ^^s. 

ANOTHER   SOLUTION. 

A  pound  of  tea,  in  the  barter,  will  buy  56-4-22  =  2/y  pounds 
of  coffee;  and  this  is  worth  $.25  x  2y«y=:$.63yV  But  the 
pound  of  tea  given  is  worth  $.621;  and  $63/y~$.62i  = 
$.01^2,  the  gain  on  $.621.  Hence,  lJL^62i=.01yV,  the 
gain  per  cent. 

Ex.  91.  First  find  how  much  ready  money  will  cancel  a  debt 
of  $1,  due  in  4  equal  installments,  for  the  times  and  at  the 
rate  mentioned, 

$  .25-^-1.011  =$.245901  -f,  pres.  worth  of  1st  installment; 
$  .25-^1.0375=$.240965+,         *'  "  2d  " 

$  .25^1.05      =$.238095+,         "  «  3d  " 

$  .25^1.081   =$.230769+,         "  «  4th         " 

$1.00  $.955730  +  ,  present  worth  of  $1. 

Now  ii'  $1,  payable  as  by  the  conditions  named,  be  worth 
$.95573  in  ready  money,  what  sum  will  be  worth,  in  readj 
money,  $750,  on  the  same  conditions  ? 

$1  :  (     )  :  :  $750  :  $.95573  ;  $784.74  +  ,  Ans. 


156 


PROMISCUOUS    EXAMPLES. 


Ex.  92    |— 11=1  of  the  capital  of  either  must  be  equal  to 
$500.     Therefore  $500  x  8  =  $4000,  Ans. 


Ex.93. 


Due. 


Dec.  4 

Feb.  9 

Feb.  29 

March  4 

May  12 


da. 


00 
67 
87 
91 
160 


Items. 


240.75 
137.25 

65.64 
230.36 

36.00 


710.00 


Prod. 


9195.75 

5710.68 

20962.76 

5760.00 


41629.19 


41629.19-^710  =  59  da.,  average  term  of  credit. 
Dec.  4,  1859  +  59  da.=Feb.  1,  1860,  Arts. 

Ex.  94.  When  he  had  spent  i  of  his  fortune,  he  had  f  lelt 
I  ofi  =  /^.  He  had  spent,  in  all,  i  + 2^=4  of  his  fortune; 
consequently,  the  $2524  which  he  had  left  was  ^  of  his  for- 
tune.    $2524-^^  =  $5889.33-f ,  whole  fortune,  Ans. 

Ex.  95 >  By  the  conditions,  the  payments  consist  of  five 
several  parts  or  installments  of  the  $3000  and  interest  on  the 
first  part  or  installment  for  1  yr.,  on  the  second  installment 
for  2  yr.,  on  the  third  for  3  yr.,  on  the  fourth  for  4  yr.,  and 
on  the  fifth  for  5  yr. ;  and  the  payments,  each  of  which  con- 
sists of  the  sum  of  one  installment  and  its  interest,  are  equal 
to  each  other.  That  is,  the  amount  of  first  installment  for 
1  yr.= amount  of  second  installment  for  2  yr.= amount  of 
third  payment  for  3  yr.,  and  so  on.  But,  as  the  payments  are 
made  annually,  the  interest  must  be  added  to  the  principal  at 
the  end  of  each  year  ;  consequently  the  second  year's  interest 
is  less  than  the  first  by  the  interest  on  the  first  installment, 
and  the  second  installment  must  exceed  the  first  by  this  in- 
terest, or  by  .07  times  the  first ;  therefore,  the  second  install* 
ment  =1.07  times  the  first.  For  similar  reasons  the  third 
installment  =1.07  times  the  second,  =1.07  x  1.07  times  the 
first,  and  sa  on  to  the  last.  Hence,  the  installments  form  a 
Geometrical  Series,  of  which 


MENSURATION. 


157 


Ist  installm.=:  1st  term  ;  5,  (numb.  ofpaym'ts)=No.  of  terms; 

1.07=ratio;  and  $3000  =  sum  of  the  series. 
That  is,  we  have  the  ratio,  the  number  of  terms,  and  the  sura 
of  the  series,  to  find  the  first  term. 
$3000  X  (1.07-1'=).07_^ 

(i.ovyi^i 

1521.674  (1st  installment) +  $210  (int.  on    $3000  for  1   yr.) 
=  1^731.674,  annual  payment,  Ans, 


:1st  term=:$521.674-|-  ; 


Ex.96. 


Due. 

mo. 

Itoms. 

Prod. 

Jan.   1,  1859 
Sept.  1,     " 
Apr.  1,1860 

0 

8 

15 

200 
350 
500 

2800 
7500 

1050 

10300 

10300-7-1050  =  9  mo.  24  da. 
Jan.  1,  1859  4-9  mo.  24  da.=Oct.  24,  1859,  Ans, 

Ex,  97.  Make  the  notes  given  the  Dr.  side  of  an  account, 
and  the  note  received  the  Cr.  side ;  the  balance  will  be  the 
other  note  received,  and  the  average  maturity,  its  date. 


Br. 

Cr. 

Dae. 

da. 

Items. 

Prod. 

Due, 

da. 

Items. 

Prod. 

July  7,  1859 
Oct.  4,     " 
Feb.  20,1860 

00 

89 

228 

600 
530 
400 

47170 
91200 

Novl5,1859 

131 

730 

95630 

1530 
730 

138370 
95630 

730956301 

Balances 

800 

42740 

42740-^800  =  53  da. 

July  7-^53  da.=Aug.  29,  1859,  note  due. 


158 


MENSURATION. 


MENSURATION  OF  LINES  AJN'D  SURFACES. 
(460,   page  342.) 
Ex.  1.     3  X  12=36  inches  long;  36  x  20=720,  Ans. 
$32x198x150 


Ex.  2. 
Ex.  3. 


160 
1000x100 


=$5940,  Ans. 
=  10  A.,  Ans, 


10000 

(469,  page  343.) 

Ex.1.     20x12=240  sq.  ch. ;    240x16  =  3840  sq.  rd.  m 
iie  meadow,  requiring  3840  rain,  for  mowing. 
3840  min.  =  6  da.  4  b.,  Ans. 

Ex.  2. 

15 


\/15^— 9^=12,  the  perpendicular; 
15x12  =  180  sq.ft.,  Ans. 


13 


Ex.  1. 


Ex.  2. 


Ex.3. 


16  +  9_ 
2 

16  +  8_ 
2 

40  +  22 


(471.) 
=  12i  in.=lJj  ft,  average  width  ; 

12  xl2V  =  12i  sq.ft.,  Ans. 
=  12  in.=l  ft.,  mean  width  ;' 

1  x8  =  8  sq.  ft.,  Ans, 
X  25=115  sq.  ch.=77  A.  5  sq.  ch.,  Ans. 


(47  5.) 

Ex.  1.     ^^  X  45=3330  sq.  fl.=370  sq.  yd.,  Ans. 
Ex.  2.     The  two  ends  together  are  equal  to  a  rectangle,   2H 
feet  by  7  feet;  hence  28  x  7  =  196  sq.  ft..  Ans. 


MENSURATION.  159 

(47  9^  page  344.) 
Ex.  1.     5x3.1416  =  15.708  ft.  =  15  ft.  8.4 +  in.,  Ans. 

Ex.  2.  721  X  3.1416  =  2265.0936  rd.  =  7  mi.  25  rd.  1.54  + ft 

Ans, 
Ex.  3.     33  X  .3183  =  10.5  + yards,  Ans. 

(480,  page  344.) 

Ex.1.     11^211^=10028.15,  Ans. 
4 

Ex.  2.     Reversing  the  rule  (II,) 

1  sq.  mi.-^.7854= 1.2732 +,  square  ot  aiamet«r, 
^.2732  =  1.1284  +mi.  =  l  mi.  41  rd.  1.4 +ft. 
Ex.  3.     84' X. 07958  =  561.5 +P.  =  3  A.  81.5  +  P.,  Ans. 


k:    E    y 


MISCELLANEOUS    EXAMPLES 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC. 


97. 

2.  Since  he  sold  -^  of  his  share,he  had  |  left;  and  |  of  J  is  ^, 
Therefore,  if  a  man  owning  f  of  a  share  in  the  Central  Rail- 
road sold  -f  of  it,  he  had  |  of  a  share  left. 

3.  Since  he  gave  J  of  it  away,  he  had  |  of  it  left;  and  §  of 
\  is  i.     Therefore,  etc. 

4.  Since  he  gave  |  of  it  for  a  knife,  he  has  |  of  it  left,  >»nd 
I  of  I  is  y^^.     Therefore,  etc. 

5.  Since  $18  was  |  of  what  the  watch  cost,  he  lost  J  of  the 
cost,  which  is  i  of  $18  or  6.     Therefore,  etc. 

6.  Since  $45  was  f  of  the  cost,  he  gained  i,  which  is  J  ot 
$45,  or  $5.     Therefore,  etc. 

'7.  Since  -f  of  the  cost  was  sacrificed,  $120  is  |  of  the  cost; 
f  of  $120,  which  is  $30,  is  |,  and  3  times  $30  is  $90,  the  whole 
loss.     Therefore,  etc. 

8.  Since  he  lent  i  of  the  remainder,  $22  is  |,  and  i  of  ^22, 
which  is  $11,  is  i  ;  3  times  $11  is  $33,  or  the  whole  of  the 
remainder;  and  $33  is  J^  of  4  times  $33  which  is  $132. 
Therefore,  etc. 

9.  Since  $80  v^as  ^  of  |  of  2  times,  or  |  the  cost,  the  losa 
was  i  which  is  J  of  $80,  or  $40      Therefore,  etc. 

(US) 


162  MISCELLANEOUS   EXAMPLES   IN   THE 

10.  Since  S^54  was  ^  of  2  times,  or  -f  the  selling  price,  lie 
gained  |,  which  is  J-  of  $54,  or  $9.     Therefore,  etc. 

11.  Since  15  is  f,  ]•  of  15,  which  is  3,  is  } ;  8  times  3  's  24, 
and  i  of  24  is  8.  Therefore,  etc.  Or,  i  of  that  number  of 
which  \  of  15  is  i. 

12.  Since  4  is  |,  i  of  4,  which  is  2,  is  ^;  3  times  2  is  6, 
which  is  I  of  2  times  6,  or  12.     4  from  12  leaves   8,  aL^d 
limes  8  is  16.     Therefore,  etc. 

13.  Since  he  sold  J  of  his  flock,  20  must  be  f  ;  |  of  20, 
which  is  10,  is  ^,  and  5  times  10  is  50.     Therefore,  etc. 

14.  Since  |  of  the  remainder,  or  |  of  J  which  is  |,  was  in 
the  water,  the  3  feet  in  the  mud  must  be  the  remaining  a  ;  and 

3  feet  is  i  of  5  times  3  or  15  feet.     Therefore,  etc. 

15.  J  +  1  =  1- j,  and  the  4  years  equals  the  remaining  y^,  and 

4  years  are  /_  of  30  years.     Therefore,  etc. 

16.  Since  $20  =  1  the  cost  of  the  coat  plus  $12,  $20— $12  = 
$8,  or  I  the  cost;  I  of  $8,  or  $4  =  i,  and  3  times  $4  is  $12. 
Therefore,  etc. 

17.  If  i  the  number +  80  was  5  more  than  3  times  the  num- 
ber, 80  —  5  =  75,  when  added  to  I  the  number  must  have  been 

3  times  the  number.  3  — |,  and  J  —  i  =  f  ;  and  75  must  have 
been  f  times  the  number.  75  is  f  of  2  times  }  of  75,  or  30. 
Therefore,  etc. 

18.  Since  16  is  |,  |  of  16  or  8  is  i,  and  3  times  8  is  24, 
which  is  twice  as  many  as  James  has  ;  then  i  of  24  or  12  is  what 
James  has  after  losing  16,  and  12  +  16  =  28,  or  the  number 
James  had  at  first ;  since  J  of  John's  equaled  ^  of  James's, 
^  of  ^  or  I  of  James's=^  of  John's ;  |  of  28  is  4,  and  5  timea 

4  is  20,  or  what  John  had  at  first.     Therefore,  etc. 

19.  Since  J +  10  ycars=:l^  or  J  of  his  age,  f  —  3  — J  or  J 
must  =10  years;  and  2  times  10  years  is  20  years.  There- 
fore, etc. 

20.  Since  |  of  $40  is  /_,  i  of  |,  or  i  of  $40  is  j\  ;  i  of  $40 
is  $8,  and  11  times  $8  is  $88,  which  is  2|  or  |  times  i  ;  }  of 

(145,  146) 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC.         163 

$88  or  $11  is  i  of  3  times  §11  or  $33,  and  $33  is  i  of  4  times 
$33,  or  $132.     Therefore,  etc. 

21.  4  times  f  is  f  ;  and  25  is  f  of  4  times  i  of  25  which  is 
20.     Therefore,  etc. 

22.  Since  he  made  away  with  f  of  |  of  it,  the  $10  left  was 
the  remaining  |  of  |  or  I  of  the  whole ;  and  5  times  $10  is  $50. 
Therefore,  etc. 

23.  Since  |  of  $1500  is  f,  J  of  |,  or  j%  of  $1500,  which  is 
$200,  is  1 ;  8  times  $200  is  $1600,  which  is  4  times  the  cost 
of  the  barn,  and  J-  of  $1600  is  $400.     Therefore,  etc. 

24.  Since  |  of  500  or  300  men  was  |  of  yf ^,  or  -^^  of  the 
force,  50  times  300  men  or  15000  men,  was  the  whole  force. 
Therefore,  etc. 

25.  Since  f  of  100  or  60  was  7i  or  J/»  tj  of  60  which  is  4 
is  J ;  2  times  4  is  8,  which  is  y^^  of  100  times  8  or  800. 
Therefore,  etc. 

26.  Since  f  of  60  or  50  was  2^  or  f  times  yi^,  i  of  50  or  10 
men  is  |,  and  2  times  10  is  20,  which  is  j^q  of  150  times  20 
or  3000  men.     Therefore,  etc. 

27.  Since  f  of  100  or  80  was  1^  or  y-,  yV  of  80  or  8  is 
|.  1  times  8  is  5Q,  less  6  =  50  or  ^\.  And  20  times  50  is 
1000.     Therefore,  etc. 

28.  Since  2i  times  30  or  70  is  3i  or  V",  rV  of  70  or  1  is  i, 
3  times  7  is  21,  which  is  j\  of  10  times  21  or  210.  There- 
fore, etc. 

29.  Since  i  of  1200  or  1000  is  8^  or  V',  sV  of  1000  or  40 
is  1  ;  3  times  40  is  120,  which  is  yf^  and  i  of  120  or  20  ia 
^^0  ;  100  times  20  or  2000  is  j  of  10000,  which  lacking 
1000  of  being  the  whole  army,  10000  +  1000  =  11000.  There- 
fore,  etc. 

(146,147) 


104  MISCELLANEOUS   EXAMPLES  IN  THE 

98. 

2.  Since  ^6 +  $4  or  $10,  bouglit  40  bushels,  $1  would  bujf 
1*0  of  40  bushels  or  4  bushels,  aud  $6  would  buy  6  times 
4  bushels  or  24  bushels,  and  |4,  4  times  4  bushels  or  16 
bushels.     Therefore,  etc. 

3.  Since  they  approach  each  other  4  miles +  3  miles  or  7 
miles  an  hour,  they  will  meet  in  Y-  or  7  hours,  and  the  one  who 
traveled  4  miles  per  hour  would  travel  7  times  4  or  28  miles, 
and  he  who  traveled  3,  Y  times  3  or  21  miles.     Therefore,  etc 

4  Since  3  weeks +2  weeks  or  5  weeks'  hire  is  $25,  1  week 
IS  i  of  $25  or  $5,  2  weeks  2  times  $5  or  $10,  and  3  weeks  3 
times  $5  or  $15.     Therefore,  etc. 

5.  Since  5  cows -f  3  cows  or  8  cows'  pasture  cost  $24,  1 
cow's  pasture  cost  |  of  $24,  or  $3,  5  cows'  5  times  $3  or  $15, 
and  3  cows'  3  times  $3  or  $9.     Therefore,  etc. 

6.  2  horses  for  2  weeks=l  horse  4  weeks,  and  2  horses  4 
weeks  =  l  horse  8  weeks;  and  since  12  weeks'  pasture  cost 
$12,  1  week's  pasture  costs  y*2  of  $12,  or  $1,  4  weeks'  4  times 
$1  or  $4,  and  8  weeks'  8  times  $1  or  $8.     Therefore,  etc. 

7.  Since  9  cents +  7  cents  or  16  cents  bought  32  figs,  1  cent 
would  buy  y*g  of  32  or  2  figs,  9  cents  would  buy  9  times  2  or 
18  figs,  and  7  cents  would  buy  7  times  2  or  14  figs.  There- 
fore, etc. 

8.  A's  $10  for  5  months=:$5  for  1  month,  B's  $5  for  8 
months=:$40  for  1  month;  and  since  $50 +  $40  or  $90  gain 
$45,  $1  will  gain  J^  of  $45,  or  $i,  $50  50  times  $^,  or  $25, 
and  $40,  40  times  $i,  or  $20.     Therefore,  etc. 

9.  Since  they  paid  in  the  proportion  of  $5,  $4,  and  $3j. 
they  own  in  the  same  proportion  ;  consequently  the  gain  ii 
divided  into  5  plus  4  plus  3,  or  12  parts,  and  j\  of  $24  is  $2. 
A's  portion  is  5  times  $2  or  $10;  B's,  4  times  $2  or  $8; 
and  C's,  3  times  $2  or  §6.     Therefore,  etc. 

10.  Since  A  does  2  times  3  days,  or  6  days'  work,  B  3 
times  3  days,  or  9  days'  work,  and  C  3  times  2^  or  5  days' 

(147,  148) 


PROGRESSIVE    INTELLECTUAL    ARITHMETIC.         165 

work,  it  takes  20  days  to  mow  the  field  ;  1  day^s  work  c/jsts 
5?o  of  |40,  or  $2  ;  and  A  should  receive  6  times  $2,  or  $12; 
B  9  times  $2,  or  18  ;  and  C  5  times  $2,  or  §10.    Thei'efore,  etc. 

11.  Since  C  took  $10,  or  :^2  — -^V  of  the  gain,  he  must  have 
put  in  2^^  of  the  stock,  and  A's  $30  plus  B's  $50,  or  $80  =  |f ; 
^V  of  $80,  or  $5=^\,  and  5  times  $5  or  $25=:iC's  siock 
$42— $10=$32,A's  +  B'sgain;  A's=ff  or  f  of  $32,  or  $12; 
end  B'sr=f  J  or  ^  of  $32,  or  $20.     Therefore,  etc. 

12.  Since  he  put  in  f^J  or  f  of  the  capital,  he  should  also 
lake  f  of  the  gain;  f  of  $240rr$150,  and  $150-$145=$5 
loss.     Therefore,  etc. 

13.  Since  2  colts  consume  as  much  as  3  calves,  4  colts,  or  2 
times  2  colts=:2  times  3  calves,  or  6  calves,  and  5  calves  plus 
6  calves  =  ll  calves.  If  11  calves  cost  $11,  1  calf  cost  j\  of 
$11,  or  $1 ;  5  calves  5  times  $1,  or  $5;  and  6  calves  6  times 
$1,  or  $6.     Therefore,  etc. 

14.  Since  C  pays  |-  of  the  rent,  he  puts  in  i  of  the  cows. 
Then  A's  5  cows  +  B's  3  cows =8  cows--=|  of  the  cows,  and 
|-  of  8,  or  4  cowsr^C's  number.  And  since  C's  4  cows  cost  i 
of  $42  or  $14,  1  cow  cost  {  of  $14,  or  $3i ;  5  cows  cost  6 
times  $31,  or  $17^;  and  3  cows  3  times  $31,  or  $10^. 
Therefore,  etc. 

15.  Since  4  cows=3  oxen,  8  cows,  being  twice  4,  =  2x3, 
or  6  oxen  ;  and  since  5  calves  =  4  cows,  10  calves,  being  twice 
6,  =2  X  4  or  8  cows.  But  8  cows=:6  oxen;  and  9  oxen +  6 
oxen -f- 6  oxen==21  oxen,  which  cost  $5B.  1  ox  cost  ^^  of 
$5G  or  $2| ;  9  oxen  9  times  $2|,  or  $24  ;  and  6  oxen  6  times 
$2 1,  or  16  ;  etc. 

16.  Since  Mary  wrote  |  as  many  lines  as  Melissa,  Melissa's 
work  IS  divided  into  8  parts,  7  of  which =Mary's;  then  8  +  7 
=  15 ;  and  j\  of  60  is  4  ;  j\  8  times  4,  or  32 ;  and  y''^  7  times 
4,  or  28.     Therefore,  etc. 

17.  Since  the  boys  received  as  many  pears  as  the  girls, 
tbey  received  i  of  24,  or  12.     There  were  as  many  boys  as 

8.R.P.       '      (148,  149) 


166  MISCELLANEOUS  EXAMPLES   IN   THE 

}  is  contained  tioies  in  12,  whicli  is  4  times;  as  many  girls 
is  4  is  contained  times  in  12,  which  is  3  times ;  and  4  +  3  =  7, 
rherefore,  etc. 

18.  Since  each  son  received  ^  as  much  as  each  daughter,  the 
2  sons  received  as  much  as  1  daughter ;  then  we  have  $96 
divided  into  3  +  1=4  parts;  }  of  $96=$24=each  daughter's 
portion ;  and  J  of  $24=$12=each  son's  portion.  Therefore,  etc. 

99. 

1  The  1st  has  1  part,  the  2d  1  part  +2,  and  the  3d  1 
part-T'2  +  6;  then  3  parts  + 2 +  2 +  6  =  76,  or  76  =  3  parts 
+  10;  and  76  —  10,  or  66  =  3  parts;  J  of  66  or  22=what 
1st  boy  had  ;  22  +  2,  or  24  =  what  2d  boy  had  ;  and  22  +  2  +  6, 
or  30  =  what  3d  boy  had.     Therefore,  etc. 

2.  Henry  has  2  more  than  James,  and  Joseph  having  2 
more  than  Henry,  has  4  more  than  James ;  hence  72,  the 
sum  of  all,  is  2  +  4,  or  6  more  than  if  each  had  no  more 
than  James.     l2  —  Q  =  G6yAns, 

3.  If  Henry  had  2  more  he  would  have  as  many  as  Joseph ; 
and  James  +  2=Henry,  and  +2  more= Joseph  ;  and  72  +  2 
+  2  +  2  =  78,  Ans. 

4.  If  Joseph  give  James  2,  Joseph's  number  will  be  dimin- 
ished and  James's  increased  2,  when  each  will=Henry's. 
James  will  now  have  ^  of  72  —  6,  or  22  ;  Henry  22  +  2,  or  24  ; 
and  Joseph  24  +  2,  or  26.     Therefore,  etc. 

5.  Since  C  paid  as  much  as  A  and  B,  he  paid  ^  of  $600, 
or  $300;  and  B  and  A  paid  $300.  And  as  B  paid  $100 
more  than  A,  $300  — $100==$200,  or  what  each  would  have 
paid  if  they  had  paid  no  'nore  than  A.  J  of  $200  is  $100, 
or  what  A  paid  ;  and  $100  +  $100  =  $200,  what  B  paid. 

6.  The  drum  cost  T  part,  the  rifle  twice  as  much,  or  2 
parts.,  and  the  watch  twice  as  much  as  the  rifle,  or  4  parts ; 
hence  $42  is  divided  into  1  part +  2  parts +  4  parts,  or  7 
parts.  I  of  $42,  or  $6  =  costof  drum;  2  times  $6,  or  $12  = 
cost  of  rifle;  and  2  trmes  $12,  or  $24 —cost  of  watch 
Therefore,  etc.  (149) 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC.         167 

Y.  The  harness  cost  2  parts,  the  horses  4  times  as  much 
or  8  parts,  and  the  wagon  IJ  times  the  harness,  or  3  parts; 
and  the  harness  2  parts  4- the  horses  8  parts,  plus  the  wagon 
5  parts=15  parts=$225.  j\  of  $225  is  $15,  and  2  times 
$]5,  or  $30==harness  ;  8  times  $15,  or  $120=:cost  of  horses ; 
and  5  times  $15,  or  $75z=cost  of  wagon,  etc. 

8.  Since  he  traveled  ^  as  far  the  1st  as  the  last  2  daya, 
the  last  2  days'  travel  is  divided  into  2  parts  ot  which  1  = 
first  day ;  hence  ^  of  114  miles,  or  38  milesi^lst  day;  the 
same  reasoning  applied  to  the  last  day  gives  38  miles,  and 
leaves  38  miles  for  the  2d  day. 

10.  The  note  of  $20  was  less  than  ^  of  what  remained 
due  after  the  1st  payment,  by  the  $20  that  exceeded  \  ;  hence 
$20 +  $20,  or  $40=r|.  2  times  $40,  or  $80  =  what  remained 
after  1st  payment,  and  $80  was  less  than  ^  the  debt,  by  the 
$10  the  payment  exceeded  i ;  $80 +  $10,  or  $90= J  ;  and  2 
times  $90,  or  $180= the  whole  debt. 

11.  The  4  pennies  left  is  less  than  \  of  the  remainder  by 
the  1  penny  more  than  ^  paid  for  the  whip;  then  4+1=5 
pennies,  or  ^,  and  2  times  5,  or  10  pennies = remainder  after 
purchasing  top ;  and  since  he  paid  2  pennies  more  than  \ 
of  all  for  the  top,  10  pennies  +  2  pennies,  or  12  peunies=| ; 
^  of  12  or  6  pennies=i  ;  and  3  times  6  =  18  pennies.  There- 
fore, etc. 

12.  Since  he  sold  the  whole,  the  \  gallon  more  than  |  the 
remainder  sold  was  ^  of  the  remainder,  and  2  times  ^  or  1 
gallon = remainder.  The  gallon  left  after  1st  sale  was  less 
than  1  the  keg  by  the  |  gallon  more  than  the  \  gallon  sold  ; 
then  1  gallon +  i  gallon  or  1|  gallons  =  ^  the  keg,  and  2 
times  1 1^  or  3  gallons=the  contents  of  the  keg. 

14.  Since  |  of  John's  =  J  of  Mary's,  \  of  John's=-J-  of  J  or 
f  of  Mary's,  and  |  or  all  of  John's =3  times  f  or  |  of  Mary's ; 
hence  xMary's  are  divided  into  8ths  and  John's =9  of  them, 
and  the  whole = J/  of  Mary's,  y^y  of  34  or  2  is  | ;  9  times  2 
or  18= John's,  and  8  times  2  =  16  =  Mary's. 
(1195  150) 


168  MISCELLANEOUS   EXAMPLES   IN    THK 

15.  Since  |  of  A's  plus  8=B's,  B's— 8  =  |  of  A's;  and  if  8 
be  taken  from  B's,  the  sum  of  both  flocks  will  be  83  —  8  or  75. 
A  has  3  parts,  B  2,  and  both  5.  ^  of  75  is  15.  3  times  15 
or  45=  A's;  and  2  times  15  or  30, +  8  =  38,  B's. 

16.  Since  f  of  Mary's  less  10  cents = Susan's,  Susan's  f  10 
cents=:f  of  Mary's,  and  then  both  would  have  39  +  10  or  49 
cents.  Mary  having  4  parts,  and  Susan  3,  they  both  have  7 
parts.  I  of  49  or  7  =  1  part;  4  times  7  or  28=Mary's;  and 
3  times  7  or  21  — 10  =  11  =:Susan's. 

17.  Since  J  of  Homer's=-f  of  Silas's,  J  of  Homer's  will=J- 
of  f  or  f  of  Silas's,  and  f  or  the  whole  of  Homer's,  5  times  f 
or  Jy"  of  Silas's;  and  since  Homer's  exceeds  Silas's  by  ^  of 
Silas's,  the  3  marbles  must=^  of  Silas's ;  hence  Silas  has  1 
marbles  and  Homer  10. 

100. 

1.  Since  the  first  drink  a  gallon  in  8  days,  he  will  drink  i 
of  a  gallon  in  1  day,  and  since  the  second  drink  a  gallon  in  4 
days,  he  will  drink  J  of  a  gallon  in  1  day  ;  both  will  drink 
i+i  or  -{fj  of  a  gallon  in  1  day,  and  1  gallon  will  last  as  many 
days  as  y'g,  what  they  drink  in  1  day,  is  contained  times  in  j| 
or  1  gallon  ;  y\  is  in  y|  1-f  times.     Therefore,  etc. 

2.  Since  Julia  can  do  it  in  7  hours,  in  4  hours  she  can  do  4 
of  it,  and  Jane  must  do  the  remaining  ^  ;  and  since  Jane  does 
if  in  4  hours,  she  will  do  |  in  i-  of  4  or  li  hours,  and  ^,  or  tht 
whole,  in  7  times  li,  or  9^  hours.     Therefore,  etc. 

3.  Since  the  first  can  do  it  in  9  hours,  he  can  do  f  in  5 
hours,  and  the  second  must  do  the  remaining  |- ;  and  since  the 
second  pitches  |^  in  5  hours,  he  can  pitch  J  in  |  of  5,  or  1 1 
hours,  and  f,  or  the  whole,  in  9  times  1^,  or  11;  hours. 

4.  3f=Y-  and  7{=Y-.  Since  the  second  pipe  can  empty 
t  in  \^  hours,  it  can  empty  j\  of  it  in  |,  and  |f  in  Y"  l^ours, 
and  the  first  must  empty  the  remaining  || ;  and  since  the  first 
can  empty  ||  in  ^■^-  hours,  it  can  empty  -^\  in  gV  of  W  or  j 
hour,  and  |-|,  or  the  whole,  in  58  times  |,  or  7|  hours. 

(151) 


PROGRESSIVE   INTELLECTUAL   ARITHMEIKJ.         169 

6.  Since  A  can  make  a  vest  in  |  of  a  day,  he  can  make 
as  many  vests  in  a  day  as  f  is  contained  times  in  |,  or  1^ 
vests ;  B  as  many  as  f  is  contained  times  in  J,  or  1-^  vests  ; 
and  l|  +  li,  or  3  vests=:what  A  and  B  can  both  do.  C  can 
make  as  many  as  f  is  contained  times  in  |,  or  1}  vests,  and 
3— li  =  l|.     Therefore,  etc. 

6.  Susan  can  knit  as  many  pairs  as  |  is  contained  timei 
m  I,  or  1|  pairs;  Sarah  can  knit  as  many  as  -3  is  contained 
times  in  ^,  or  2i  pairs;  and  l|  +  2^  =  4  pairs. 

7.  Since  Sarah  can  knit  2i  or  J  pairs  in  a  day,  she  can 
knit  i  of  a  pair  in  |  of  a  day,  which  is  the  part  she  must 
knit  for  Susan. 

8.  Since  Susan  can  knit  1|  or  f  pairs  in  a  day,  she  can 
knit  ^  of  a  pair  in  i  of  a  day,  which  is  the  part  she  must  knit 
for  Sarah. 

9.  Since  Jason  can  hoe  10  rows  in  J  of  an  hour,  he  can 
hoe  1  row  in  y^  of  J,  or  j%  of  an  hour,  and  as  many  rows 
in  an  hour  as  3  is  contained  times  in  40  or  131  rows. 
Since  Jesse  can  hoe  10  rows  in  j  of  an  hour,  he  can  hoe  1 
row  in  y'^  of  f  or  j\  of  an  hour,  and  as  many  rows  in  an 
hour  as  3  is  contained  times  in  50,  or  16|  rows;  and 
both  can  hoe  13i  +  16|,  or  30  rows,  in  an  hour;  1  row  in 
^^  of  an  hour ;  and  10  rows  in  i§  or  ^  of  an  hour. 

10.  Smce  Jesse  can  hoe  16|  or  \o  rows  in  an  hour,  in 
I  of  an  hour  he  can  hoe  |  of  */  or  V=^3  rows;  leaving 
1|  rows  for  Jason,  who  can  hoe  13i  or  ^«  rows  in  an  hour, 
1^  of  a  row  in  y^^  of  an  hour,  and  1|  or  f  rows  in  5  times  yV 
or  I  of  an  hour. 

11.  Since  Jason  can  hoe  13^  or  ^^^  rows  in  an  hour,  in  J  of 
an  hour  he  can  hoe  ^  of  */,  or  V=4f  rows ;  leaving  5f  rowa 
for  Jesse,  who  can  hoe  16|,  or  Y=-^p  rows  in  an  hour,  J  of 
a  row  in  yJ^  of  an  hour,  and  \%  or  5f  rows  in  50  timea  ,  jj 
of  an  hour. 

12.  See  analysis  of  Example  9. 

13.  Since  A  and  B  can  clear  the  field  in  15  days,  thev  can 

(151.  152) 


170  MISCELLANEOUS  EXAMPLES  IN  THE 

clear  f^  of  it  in  1  day,  and  -fj  or  J  of  it  in  9  days ;  and  sinco 
A  and  B  clear  |  of  it  in  9  days,  C  must  clear  the  remaining 
}  ;  and  if  he  clear  f  in  9  days,  he  will  clear  i  in  ^  of  9  or  4j 
days,  and  f  or  the  whole  field  in  5  times  4^  or  22|  days. 

14.  Since  A  and  B  can  dig  it  in  6  days,  they  can  dig  }  of 
it  in  1  day  ;  since  A  and  C  can  dig  it  in  8  days,  they  can  dig 
J  of  it  in  1  day ;  and  |— J  or  j\  of  it,  is  what  B  does  more 
in  a  day  than  C.  As  B  and  C  dig  it  in  9  days,  they  can 
di^  i  ^^  it  in  1  day,  and  since  B's  day's  work  exceeds  C's  by 
^■f  of  the  well,  J  — aV  ^^  1^=^  ^^  ^'®  ^^7^  ^.nd  J  of  yV  ^^  yf  j 
=what  C  can  do  in  1  day  ;  hence  C  can  do  it  in  as  many  days 
as  5  is  contained  times  in  144  or  28|^  days.  Since  B  and  G 
dig  I  of  it  in  1  day,  and  C  digs  yf  y  of  it  in  1  day,  i  — yf  4-  or 
yU_=what  B  digs;  hence  B  can  dig  it  in  as  many  days  as  11 
is  contained  times  in  144  or  13tV  davs.  Since  A  and  B  dis 
^  of  it  in  1  day,  and  B  digs  jW  of  it  in  1  day,  J  — yVv  or  tV\ 
=:what  A  digs  in  1  day ;  hence  A  can  dig  it  in  as  many  days 
as  13  is  contained  times  in  144  or  llyj  days. 

15.  Since  A  digs  y^^^,  B  y^^,  and  C  yf y  of  it  in  1  day 
they  will  all  dig  yW  +  yW  -f  if??  or  y^y  of  it  in  1  day  ;  and 
it  will  take  as  many  days  as  29  is  contained  times  in  144  01 
m  <^ays. 

16.  Since  Patrick  and  Peter  can  dig  it  in  15  days,  they 
can  dig  y'y  of  it  in  1  day,  and  |{  or  |  in  10  days,  and  Philc 
must  dig  the  remaining  third ;  and  since  Philo  digs  ^  in  10 
days,  he  can  dig  f  or  the  whole  in  3  times  10  or  30  days. 
Since  Philo  can  dig  it  in  30  or  */■  days,  he  can  dig  ^V  of  it 
»n  1  of  a  day,  and  in  13^  or  ^/  days  he  can  dig  40  times  ■^\ 
or  |-  of  it,  and  Peter  must  dig  the  remaining  f  ;  and  since  he 
digs  f  in  4Ji  days,  he  will  dig  |  in  ]^  of  V"  or  f  days,  and  f  or 
the  whole  in  9  times  f  or  24  days.  Since  Peter  can  dig  it 
in  24  days,  in  15  days  he  can  dig  ^f  or  f  of  it,  and  Patrick 
must  dig  the  remaining  | ;  and  since  he  digs  |  in  15  daysi 
be  will  dig  |  in  i^  of  15  or  5  days,  and  f  in  8  times  5  oz 
40  days.     As  Patrick  can  dig  40  rods  in  24  days,  he  can  dig 

(152) 


PROGRESSIVE  INTELLECTUAL  ARITHMETIC.  171 

iV  of  40  or  1|  rods  in  1  day,  and  since  Peter  ,ian  dig  40 
rods  in  40  days,  lie  can  dig  1  rod  a  day,  and  it  will  take  liirn  as 
many  days  asl|-fl  =  2|is  contained  times  in  28,  or  10|^  days. 

17.  Since  30  rods  is  JJ  or  J  of  40  rods,  it  will  take  each 
man  J  as  long  to  dig  it.  Since  Patrick  could  dig  it  in  40  days, 
he  can  dig  30  rods  in  J  of  40  or  30  days ;  since  Peter  can 
dig  it  in  24  days,  he  can  dig  30  rods  in  f  of  24  or  18  days 
and  since  Philo  can  dig  it  in  30  days,  he  can  dig  30  reds  in 
I  of  30  or  22i  days. 

18.  Henry's  work  is  divided  into  4  equal  parts,  and  since 
Harlan's  exceeds  Henry's  by  1  of  these  parts,  he  must  do  5 
parts,  and  both  of  them  4  +  5  or  9  parts.  Since  Henry  cuts 
f  of  it  in  6|  or  ^^  days,  he  can  cut  ^  in  ^  of  -^-  or  f  days, 
and  f  in  9  times  |  or  15  days.  Since  Harlan  cuts  f  of  it  in 
6|  or  2-0  days,  he  can  cut  |^  in  ]^  of  ^/-  or  ^  days,  and  f  in  9 
times  1^  or  12  days. 

19.  Since  the  3d  does  |  as  much  as  the  1st  and  2d,  the 
work  of  the  1st  and  2d  is  divided  into  5  parts;  and  since 
the  3d  does  |  as  much,. the  whole  is  divided  into  5-f  2  or 
7  parts.  Since  the  3d  does  f  of  the  whole  in  10  days,  he 
can  do  I  in  |-  of  10  or  5  days,  and  -J  in  7  times  5,  or  35 
da;'s.  Since  the  1st  and  2d  do  ^  in  10  days,  they  can  do 
4  in  }  of  10  or  2  days,  and  ^  in  7  times  2  or  14  days.  And 
since  the  1st  does  J  as  much  as  the  2d,  the  whole  is  divided 

hto  7  parts,  of  which  the  1st  does  3,  and  the  2d  4  parts. 
Smce  the  1st  does  ^  in  14  days,  he  can  do  |  in  |^  of  14  or  4 1 
days,  and  ^  in  7  times  4|  or  32|  days.  Since  the  2d  does  \ 
in  14  days,  he  can  do  |  in  {^  of  14  or  3^  days,  and  ^  in  7 
times  3i  or  24^  days. 

20.  See  Analysis  of  example  19. 

21.  Since  the  1st  can  do  it  in  32|  or  *3^  days,  he  can  do 
j'j  of  it  in  |-  of  a  day  or  -^j  in  a  day ;  and  since  the  3d  cau 
do  it  in  35  days,  he  can  do  j'j  of  it  in  1  day ;  and  both  can  do 
^8  +  3  J  ^^  -Ns  ^^  1  ^^y»  *^^  ^^^  whole  in  as  many  days  ag 
29  is  contained  times  in  490  or  l^-^j  days. 

(152) 


172  MISCELLANEOUS  EXAMPLES  IN   THE 

22.  Since  the  2d  can  do  it  in  24^  or  Y"  ^^ys,  he  can  dc  j^ 
of  it  in  i  of  a  day,  or  /^  in  a  day ;  and  since  the  3d  can  io  ^^-g 
of  it  in  1  day,  the  2d  and  3d  can  do  /g +-3J,  or  g'^^  of  it  in  1 
day,  and  they  can  do  all  of  it  in  as  many  days  as  17  is  con- 
tained times  in  245,  or  14^'^^  days. 

23.  Since  B  and  C  can  do  it  in  12  days,  they  can  do  y"^  cr 
I  of  it  in  8  days,  and  A  must  do  the  other  i ;  and  since  A  can 
do  "I"  in  8  days,  he  can  do  ^  in  3  times  8  or  24  days.  Since  A 
and  B  can  do  it  in  10  days,  they  can  do  j%  or  J  of  it  in  8 
days,  and  C  mast  do  the  other  \  ;  and  since  C  can  do  |  in  8 
days,  he  can  do  |  in  5  times  8,  or  40  days.  Since  A  can  do 
it  in  24  days,  he  can  do  i|  or  /^  of  it  in  10  days,  and  B  must 
do  the  remaining  y^g ;  and  since  B  can  do  y\  i^  ^^  ^^J^i  ^g  can 
do  j\  in  I  of  10  or  1^  days,  and  -ff  in  12  times  If  or  17 j  dayt. 

24.  Since  the  1st  and  2d  will  discharge  it  in  8  hours,  they 
'discharge  f  or  i  of  it  in  4  hours,  and  the  3d  must  discharge 
the  other  ^ ;  and  since  it  discharges  -^  in  4  hours,  it  will  dis- 
charge I  in  2  times  4,  or  8  hours.  Since  the  3d  will  discharge 
it  in  8  hours,  it  discharges  f  or  |  of  it  in  6  hours,  and  tho 
1st  must  discharge  the  other  \  ;  and  since  the  1st  discharges 
}  of  it  in  6  hours,  it  will  discharge  |  in  4  times  6  or  24  hours. 
Since  the  1st  and  3d  discharge  it  in  6  hours,  they  will  dis- 
charge f  or  I  of  it  in  4  hours,  and  the  2d  must  discharge  the 
other  i  ;  and  since  the  2d  discharges  J  in  4  hours,  it  will  dis^- 
charge  f  in  3  times  4  or  12  hours. 

25.  Since  A  and  B  can  do  it  in  20  days,  they  do  ij  or  f 
of  it  in  10  days,  and  C  does  the  other  i  ;  and  since  C  does  | 
in  10  days,  he  can  do  |  in  2  times  10  or  20  days.  Since  B 
and  C  can  do  it  in  15  days,  they  do  j|  or  |  of  it  in  10  days, 
and  A  does  the  other  third  ;  and  since  A  does  ^  in  10  days, 
he  can  do  f  in  3  times  10  or  30  days.  Since  A  can  do  it  in 
80  days  and  C  in  20  days,  they  can  both  do  3V  +  2V  or  y'^  of 
it  in  1  day,  and  if  in  12  times  1  or  12  days. 

26.  Since  it  would  last  them  aH  30  days,  they  would  eat  3'^ 
of  it  in  1  day,  and  20  times  gV  or  |  of  it  in  20  days,  leaving  \ 

(152,  153) 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC.         173 

of  it  to  be  eaten  by  the  sister.  Since  the  brother  and  servant 
would  eat  it  in  45  days,  they  would  eat  |f  or  |  of  it  in  30 
days,  and  the  sister  must  eat  the  other  J  in  30  days. 

101. 

2.  Since  2  plums  was  tbe  increase  given  to  1  playmate,  and 
9—1  or  8  plums  the  increase  given  to  all,  there  were  as  many 
!d\  ay  mates  as  2  is  contained  times  in  8,  which  is  4  times. 
Therefore,  etc. 

3.  Since  the  difference  between  6  times  and  3  times  a  num- 
ber is  3  times  the  number,  15  must  be  3  times  the  number,  and 
J-  of  15,  or  5  must  be  the  number.     Therefore,  etc. 

4.  Since  the  difference  per  yard  was  12  cents  — 8  cents,  oi 
4  cents,  she  wanted  as  many  yards  as  4  is  contained  times  in 
the  whole  difference,  11  cents +  17  cents,  or  28  cents,  which  is 
7  times. 

5.  Since  the  difference  between  6^  times  and  4  times  a 
number  is  21  times  or  f  times  the  number,  15  must  be  | 
times  the  number;  J  of  15  or  3,  i ;  and  6,  the  number. 

6.  Since  |  — i  =  |,  i  of  4  or  2  must  be  |  and  9  times  2  or 
,18  =  t.    •       • 

7.  Since  the  difference  between  5^  times  and  3|  times  a 
number  is  2.1  times  the  number,  -^j  of  21,  or  1,  must  be  y^, 
and  10  times  1,  or  10,  |J,  or  the  number. 

9.  If  we  let  1  or  |  represent  the  whole  number  of  chickens, 
I  times  5  +  J-  times  3=-'/,  will  represent  the  whole  number  of 
grains,  that  is,  "3^=26.  And  since  26  is  -y-  times  the  number 
of  chickens,  |  or  the  whole  number  of  chickens,  was  3  times  | 
of  26,  or  6. 

10.  Since  26  is  5  times  |  +  3  times  ^,  or  ^-f  of  the  number; 
}^  of  26  or  2  is  ^,  and  3  times  2  or  6  is  f  or  the  number. 
Therefore,  etc. 

11.  Since  the  1st  condition  gives  5  times  ^  of  a  number, 
plus  the  2d  condition,  which  gives  3  times  J  of  the  same  num- 
ber, plus  the  3d,  which  gives  2  times  |  of  the  same  n amber, 

(153,  151) 


174  MISCELLANEOUS   EXAMPLES  IN   THK 

plus  the  4th,  which  gives  once  J  the  same  number  ;  we  have 
^  of  the  class  equal  to  29,  and  ^^^  of  29  or  1  is  ^  of  9  times 
1  or  9.     Therefore,  etc. 

13.  Since  4  times  ^  of  a  number,  plus  S^  times  the  num- 
ber, or  -y-  times  the  number,  is  equal  to  28  plus  5,  or  33,  f\ 
of  33,  or  3  is  ^  of  the  number,  and  2  times  3  or  6  is  the  num- 
ber.    Therefore,  etc. 

14.  Since  the  1st  condition  gives  ^  of  his  age  plus  4,  the 
2d  gives  f,  and  the  3d  gives  ^  less  4,  we  have  the  sum  -'J 
equal  to  50 ;  y^  ^^  ^^  ^^  5  is  ^  of  his  age,  and  3  times  5  or 
15  is  J.     Therefore,  etc. 

15.  Since  he  paid  $5  a  head  for  ^  of  the  flock,  the  cost=:5 
times  ^  or  1^  of  his  flock;  $4  a  head  for  as  many  more  =  4 
times  I  or  J  of  his  flock;  $3  a  head  for  |  of  the  remainder, 
or  ^  =  3  times  J  or  i  of  his  flock  ;  and  82  a  head  for  the  rest, 
or  J-  =  2  times  |  or  i-  of  his  flock;  and  f  +  J  +  i  +  i  =  V' 
That  is,  the  number  of  dollars  the  flock  cost=:\3  of  f}^Q  num- 
ber of  sheep,  and  115  is  ^  o^  ^  times  gV  of  115  which  is  30 
Therefore,  etc. 

16.  Since  he  received  6  dimes 

each  for  I =1x6=1 

for  ^  of  the  remaining  |  and  3  more, 

4  dimes  each =(f +  3)  x4=f +  12dime» 

for  1  of  the  rest  (which  is  f  less  1 

on  each  eight),  or  1  +  1 =  (}  +  l)x3=:f+  3     ** 

for  the  rest  (which  is  f  less  1  on 

each  eight,  also  the  2  of  last  sale), 

or  f  less  4 =(|— 4)  x  2=| — J     ** 

the  whole  number  of  dimes  is  equal 

to  -5/  of  the  baskets  and  7  dimes  besides, . .  =-\^+   7     ** 

hence  $10  or  100  dimes  less  7  dimes=93  dimes=3_i^  _i_  qj 

93  or  3  =  |-,  and  8  eighths,  8  times  3  or  24.     Therefore,  etc. 

I      18.  6  times  a  number  equals  \^y  Y  times  J  of  it  plus  5 

dmes  1  of  it  equals  ^/,  and  Y  less  Y-  =  J  or  ^  of  it,  which, 

(154,  155) 


PROQKESSIVB   INTELLECTUAL   ARITHMETIC.        175 

according  to  the  condftion  of  the  question,  is  4 ;  and  4  is  | 
of  2  times  4  or  8.     Therefore,  etc. 

19.  6  times  the  number,  or  Y»  left  4  cents,  but  5  times  J  oi 
it,  or  y ,  plus  7  times  ^  oi  it,  or  Y-  was  it  all  of  it ;  and  by  the 
condition  of  the  question  ^^  less  y  or  \  equals  4 ;  and  f  is  2 
times  4  or  8.     Therefore,  etc. 

20.  4  times  a  number  equals  Y,  5  times  4  of  it  equals  y, 
and  Y  less  Sy^-  equals  f  of  it,  which-by  the  question  is  6 ;  an.l 
6  is  I  of  7  times  ^  of  6,  which  is  14.     Therefore,  etc. 

21.  2  times  a  number  equals  |  of  it,  5  times  i  of  it  equals 
I,  and  this  plus  2  times  \  of  it — which  is  | — equals  |  of  it, 
and  J  less  f ,  equals  J  of  it,  which  by  the  conditions  of  the 
question  is  60  ;  and  60  is  |  of  2  times  -J-  of  60  which  is  40. 
Therefore,  etc. 

22.  2  times  }  of  a  number  equals  f  of  it,  which  is  8  more 
than  J'  hence  8  is  f  or  i  of  it,  and  2  times  8  or  16  is  the 
whole  of  it     Therefore  etc, 

103. 

2.  Since  19  is  the  sum  of  two  numbers  whose  difference  is 
8, 19  less  3,  or  16,  is  twice  the  less  number;  ^  of  16  is  8,  the 
less  number,  which,  increased  by  3,  equals  11,  the  greater 
number.     Therefore,  etc. 

3.  Since  31  is  the  sum  of  two  numbers  whose  difference  is 
9,  31  less  9  or  22,  is  twice  the  less  number ;  |  of  22or  11  is* 
the  less  number,  which,  increased  by  9,  equals  20,  the  greater 
number.     Therefore,  etc. 

4.  Since  37|^  is  the  sum  of  two  numbers  whose  difference  it 
5^,  3Y|  less  5^  or  32,  is  twice  the  less  number ;  |  of  32  is  16 
the  less  number,  which,  increased  by  5^,  equals  21J-,  the 
greater  number.     Therefore,  etc. 

5.  Since  21  is  the  sum  of  two  numbers  whose  difference  is 
5,  21  less  5  or  16,  is  twice  the  less  number;  |^  of  16  is  8; 
the  number  Homer  had  at  first,  plus  3,  equals  11,  or  what 

(155,  156) 


176  MISCELLANEOUS   EXAMPLES  IN   THE 

he  lias  now  ;  and  21  less  11,  or  10,  equals  what  Horace  h^ 
now.     Therefore,  etc. 

6.  Since  Mary  has  twice  as  many  as  Martha,  she  has  2 
parts,  and  Martha  1,  they  both  have  3  parts;  ^  of  12  quarts  or 
4  quarts,  equals  what  Martha  has,  and  twice  4  or  8  quarto 
equals  what  Mary  has.     Therefore,  etc. 

7.  Since  47  is  the  sum  of  two  numbers,  one  of  which  is  S 
nore  than  twice  the  other,  47  less  5,  or  42,  equals  3  times 
he  less  number ;  i  of  42  or  14,  equals  the  less,  and  twice  14 

or  28  plus  5,  which  is  33,  equals  the  greater.     Therefore,  etc. 

8.  If  the  small  bin  held  6  bushels  more,  it  would  contain  J 
as  much  as  the  other,  and  both  would  hold  60  bushels,  or  3 
times  as  much  as  the  small  one ;  ^  of  60  or  20,  less  6,  which 
is  14,  equals  the  number  of  bushels  in  the  smaller  bin,  and  2 
times  20  or  40,  equals  the  number  in  the  larger  bin.  There- 
fore, etc. 

9.  Had  the  watch  cost  $4  more,  both  would  have  cost  $100, 
or  4  times  the  cost  of  the  chajn  ;  J-  of  $100,  or  $25,  equals 
the  cost  of  the  chain,  and  $96  less  $25,  or  $71,  equals  the 
cost  of  the  watch.     Therefore,  etc. 

10.  Since  Hiram  received  11  times  2,  or  22  dimes  more 
than  Harvey,  253  dimes,  what  both  received,  less  22  dimes,  or 
231  dimes,  equals  twice  what  Harvey  received;  |  of  231,  or 
1151  dimes   equals  what   Harvey  received,  and  115^  dimes, 

■plus  22  dimes,  or  137|^  dimes  equals  what  Hiram  received; 
y'y  of  115  J  dimes,  which  is  $1.05,  equals  what  Harvey  received 
per  day;  and  $1.05,  increased  by  2  dimes,  equals  $1.25,  what 
at  Hiram  received.     Therefore,  etc. 

11.  Since  B's  age  was  2  times  A's  6  years  since,  48  years^ 
the  sum  of  their  ages  then,  must  have  been  4  times  A's  age  ; 
I  of  48,  which  is  12,  plus  6,  or  18  years,  equals  A's  age ;  and 
60  less  18;  or  42  years,  equals  B's  age.    Therefore,  etc. 

12.  Since  the  horse  cost  $4  more  than  3  times  the  cost  of 
the  cow,  $124  less  $4,  or  $121,  is  4  times  the  cost  of  the 
cow;    \  of  $121,  or    $30.25,   equals  the  cost   of   the   cow 

(156,  157) 


PROGRESSIVE  INTELLECTUAL  ARITHMETIC.  177 

and  $124  less  $30.25,  or  $93.75,  equals  the  cost  of  the  lorso 
Therefore,  etc. 

13.  Since  the  product  is  the  same  whichever  factor  be  taken 
for  the  multiplicand,  we  will  use  }  of  the  cost  of  the  cow, 
which  taken  4  times,  equals  J  or  the  whole  cost ;  hence,  i  of 
the  cost  of  the  colt  must  be  $4  ;  twice  $4,  or  $8,  equals  what 
he  paid  for  the  colt;  and  $24  less  $8,  or  $16,  equals  what 
he  paid  for  the  cow.     Therefore,  etc. 

14.  Since  the  cost  of  the  cover  (which,  by  a  condition  of 
the  question  is  ^  as  much  as  the  dish  plus  the  difference), 
increased  by  the  difference,  equals  the  cost  of  the  dish,  the 
dish  costs  twice  the  difference  plus  I  of  itself,  or  the  differ- 
ence equals  J  of  the  cost  of  the  dish  ;  and  f  less  |,  or  §  of 
the  cost  of  the  dish  equals  the  cost  of  the  cover ;  and  24 
dimes  equals  |  of  the  cost  of  the  dish.  |  of  24  dimes=3 
dimes,  5  times  3  =  15  dimes,  the  cost  of  the  dish;  and  24 
dimes— 15  dimes =9  dimes,  the  cost  of  the  cover. 

15.  Since  the  less  number, — which  by  the  question  equals  | 
of  the  greater  plus  the  difference, — increased  by  the  difference 
equals  the  greater,  we  have  the  greater  equaling  -i  of  itself 
plus  twice  the  difference,  or  the  difference  equaling  j\  of  the 
gi-eater,  and  t6~"tV  — tV  ^^  ^^^  greater  equals  the  less;  |J 
4-y«g=f-f ;  2  J  of  25  pounds,  or  1  p  )und,  is  j\  of  16  pounds, 
the  greater  number,  and  9  times  1  pound  or  9  pounds  is  the 
less.     Therefore,  etc. 

16.  Since  the  sum  of  the  difference  and  the  less  number 
equals  the  greater,  the  less  must  equal  |  of  the  greater,  and 
both  numbers  f  of  the  greater ;  |  of  10  =  2  is  i  of  the  greater 
number,  3  times  2  is  6,  the  greater ;  and  10  less  6,  or  4  is  the  less, 

17.  Since  the  cost  of  ironing,  plus  ^  of  the  difference,  equals 
I*-,  of  the  cost  of  the  wood-work,  the  remaining  ^j  must  equal 
i  of  the  difference,  and  the  difference  equals  j\  of  the  cost 
of  the  wood-work ;  ||  less  y\  equals  j\  of  the  cost  of  irou- 
mg ;  II  plus  Y®,  or  |^  times  the  cost  of  the  wood-work  equali 

(157) 


178  MISCELLANEOUS    EXAMPLES    IN   THK 

|38.     i-'g  of  $38  or  $2,  is  yV»  ^  ^^^n^^s  $2  or  $22,  is  the  cost  o( 
the  wood-work,  and  $38  less  $22  or  $16  is  the  cost  of  ironing, 

18.  Since  the  cost  of  the  ribbon, — which  by  the  question 
equals  ^  of  the  cost  of  the  lace,  plus  ^  the  difference,  increased 
by  the  difference  between  the  cost  of  the  lace  and  ribbon, — 
equals  the  cost  of  the  lace ;  we  have  -^  of  the  cost  of  the  lace 
equal  to  f  of  the  difference,  or  the  lace  costing  a  sum  equal 
Ic  }  of  the  difference,  and  the  ribbon  ^  of  the  difference,  and 
both  30  cents,  or  5  times  the  difference.  |  of  30  cents,  or  0 
cents,  is  the  difference  between  the  cost  of  the  two ;  30 
cents  less  6  cents  or  24  cents,  is  twice  the  cost  of  the  ribbon, 
and  ^  of  24  cents  or  12  cents  is  the  cost  of  the  ribbon  ;  and 
30  cents  less  12  cents,  or  18  cents  is  the  cost  of  the  lace. 

19.  Since  the  whole  of  the  cost  of  the  knife  and  once  the 
difference  equals  the  cost  of  the  skates,  and  by  the  question  ^ 
the  cost  of  the  knife  plus  twice  the  difference  equals  the  same, 
once  the  difference  must  equal  ^  the  cost  of  the  knife,  twice 
the  difference  the  whole  cost,  3  times  the  difference  the  cost  of 
the  skates,  and  5  times  the  difference  equals  20  shillings,  or 
the  cost  of  both  ;  |  of  20  shillings  is  4  shillings,  2  times  4 
shillings  is  8  shillings,  the  cost  of  the  knife ;  and  3  time?*  4 
shillings  is  12  shillings,  the  cost  of  the  skates. 

20.  Had  the  harness  cost  $1  more,  both  would  have  cost  i  J5, 
and  the  horse  would  cost  ^  of  $35  or  $20,  and  the  harnesi*  ^ 
of  $35  or  $15,  less  $1  or  $14.     Therefore,  etc. 

103. 

2.  Had  all  been  old  sheep,  he  would  have  paid  $84,  or  $8 
more  than  he  did ;  each  yearling  made  a  difference  of  $1, 
hence  there  were  as  many  yearlings  as  $1  (the  difference  on 
1)  is  contained  times  in  $8  (the  difference  on  all),  which  i»  8 
times ;  and  28  less  8  equals  20,  the  number  of  old  sheep. 

3.  Had  all  been  first  quality,  he  would  have  paid  $9(  ^  ox 
$8  more  than  he  did ;  and  since  the  difference  per  barrel  Ht» 

(157,  458) 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC.         179 

$•1,  he  bought  as  many  barrels  of  poor  quality  as  $1  is  contained 
times  in  $8,  or  8  barrels;  and  20  less  8  equals  12  first  quality. 

4.  Since  he  lost  |  of  the  cost,  ^  of  $18  or  $9,  must  have 
been  J  of  the  cost,  and  5  times  $9  or  $45,  was  the  whole  cost. 

5.  There  were  as  many  of  each  as  12  dimes  (the  number 
it  took  to  pay  one  of  each)  is  contained  times  in  72  dimes 
(the  number  paid  to  all).     12  is  in  72  6  times,  and  2  times  6 

12,  the  whole  number. 

6.  Since  »he  received  8  dimes  for  1  of  each,  she  sold  as 
many  of  each  as  8  dimes  is  contained  times  in  40  dimes,  which 
is  5  times ;  twice  5  is  10,  the  number  of  fowls  she  sold. 

7.  He  bought  as  many  bushels  as  $.50,  the  difference  on  1 
bushel  of  each  is  contained  times  in  $7,  the  difference  on  all ; 
$.50  is  in  $7  14  times.     Therefore,  etc. 

9.  He  was  idle  as  many  days  as  $3.50  (the  difference  made 
by  1  idle  day)  is  contained  times  in  $7  (the  difference  made  by 
all  the  idle  days),  which  is  2  times ;  20  days  less  2  days  is  18 
days.     Therefore,  etc. 

11.  Since  she  gave  |  of  the  remainder  to  her  teacher,  the 
2  left  must  be  the  other  ^  ;  4  times  2  is  8,  which  was  the  J 
left  after  division  among  the  playmates,  and  4  times  8  or  32 
equals  the  number  she  had  at  first.     Therefore,  etc. 

12.  J  of  1 1  is  y\  ;  hence  12  is  y\  of  his  flock,  and  10  times 
12,  or  120  equals  the  number  of  sheep  he  had  at  first 

13.  Since  he  paid  |  of  the  remainder,  $3  must  be  |  of  it. 
J  of  $3  is  $1,  and  5  times  $1  is  $5,  the  remainder,  which  by 
thefirst  payment  wants  $5  of  being  i  of  the  whole ;  $5  plus 
$5  is  $10,  1  of  the  whole,  and  4  times  $10  is  $40,  the  whole 
Therefore,  etc. 

li.  Since  he  lent  |,  $3  plus  $5  or  $8,  must  have  been  i. 
8  times  $8  or  $24  is  what  he  had  after  paying  for  the  watch  ; 
$24  plus  $12  or  $36  equals  what  he  had  after  paying  for  hia 
clothes,  which  lacks  $10  of  being  i  of  his  wages;  $36  plua 
llO,  or  $46  is  ^ ;  and  2  times  $46,  or  $92  equals  his  wages. 
(158,159) 


180  MISCELLANEOUS  EXAMPLE?  IN    THE 

15.  iSince  in  $1  there  are  10  climes,  he  could  be  idle  tm 
many  days,  for  each  day  he  worked,  as  2  dimes,  what  he  paid 
a  day  for  board,  is  contained  times  in  the  amount  his  daily 
wages  exceeded  $1,  which  is  once;  hence  he  worked  ^  of  the 
time,  9vd  was  idle  10  days. 

104. 

2.  The  part  standing  was  divided  into  4  equal  parts,  3  of 
ir!iich  equaled  the  part  broken  off;  the  sum  of  both  piec^ 
uras  7  equal  parts,  1  of  which  was  |  of  56  feet  or  8  feet,  3 
parts  were  3  times  8  or  24  feet,  which  was  the  part  broken  off; 
and  4  times  8  or  32  feet  was  the  part  standing.  Therefore,  etc. 

3.  Since  Henry  has  5  parts  and  Horace  4  parts,  both  have 
9  parts ;  ^  of  45  is  5  ;  4  times  5  or  20  equals  the  number 
Hor?  ',e  had,  and  5  times  5  or  25  equals  the  number  Henry 
had.     Therefore,  etc. 

4.  Since  he  left  5  parts  and  took  out  3,  he  left  f  of  160,  or 
100  pounds.     Therefore,  etc. 

5.  Since  he  paid  5  parts  for  his  lodging  and  4  for  his  sup- 
per, his  supper  cost  J  of  63,  or  28  cents.     Therefore,  etc. 

6.  Since  9  times  |  =  |  times  the  cost  of  wagon,  equals  the 
cost  of  the  horse,  both  cost  8  plus  9,  or  y-  times  the  wagon ; 
iV  of  $170  is  $10  ;  8  times  $10,  or  ^80  was  the  cost  of  the 
wagon ;  and  9  times  $10,  or  $90  the  cost  of  the  horse. 
Therefore,  etc. 

7.  Since  the  second  day's  travel  was  1^^=^  times  the  first, 
both  equaled  f  times  the  second ;  |  of  140  miles  is  20  miles; 
3  times  20  equals  60  miles,  the  second  day's  travel;  and  4 
times  20  equals  80  miles,  the  first  day's  travel. 

8.  Since  Bergen  is  50  miles  from  Buffalo,  280  miles  less  60 
or  230  miles  equals  the  distance  from  Bergen  to  Schenectady; 
and  as  the  distance  from  Utica  to  Schenectady  is  l^zzz^j^  times 
the  distance  from  Bergen  to  Utica,  the  whole  distance  from 
Bergen  to  Schenectady  equals  y  plus  |,  or  Y  '^  _i_  of  230 
miles  is  10  miles,  and  15  times  10,  or  150  miles  equals  the 

(159,  160) 


PROGRESSIVE   INTELLECTUAL   ARITHMETIC.         181 

distance  from  Schenectady  to  Bergen  ;  and  150  miles  plus 
60  miles,  the  distance  from  Bergeu  to  Buffalo,  gives  200  miles 
from  Buffalo  to  Utica. 

9.  Since  the  head  was  3  inches  long,  17  inches  less  3  or  14 
inches  equals  the  length  of  the  body  and  the  tail;  and  as  the 
body  was  divided  into  fifth's,  2  of  which  equaled  the  tail,  we 
have  body  and  tail  divided  into  7  parts ;  |  of  14  inches  is  2 
inches,  and  2  times  2  or  4  inches  equals  the  tail. 

10.  Since  the  less  has  7  parts  and  the  greater  11,  both  have 
18  ;  jV  o^  '^6  is  2  ;  7  times  2  =  14,  the  less  part;  and  11  times 
2  =  22,  the  greater. 

12.  If  the  distance  from  Victor  to  Rochester  were  4  miles 
less,  it  would  equal  the  ^j  mentioned,  and  the  whole  distance 
would  be  52  miles  less  4  miles,  or  48  miles  ;  from  Geneva  to 
Victor  is  11  parts,  from  Victor  to  Rochester  5  parts,  in  all  16 
parts;  y^-  of  48  is  4  ;  11  times  4  miles=44  miles,  the  distance 
from  Geneva  to  Victor ;  and  52  miles  less  44  miles  =18  miles, 
the  distance  from  Rochester  to  Victor. 

13.  If  the  church  were  6  feet  lower,  the  whole  distance 
would  be  140  feet,  of  which  the  steeple  would  be  4  part's,  the 
church  3  parts,  and  both  7  parts;  |  of  140  feet  is  20  feet; 
and  4  times  20  is  80  feet,  the  height  of  the  steeple.  There- 
fore, etc. 

14.  Since  the  jar  (which,  by  a  condition  of  the  question, 
weighs  as  much  as  i  the  cover  plus  12  pounds)  and  the  cover 
weighs  18  pounds,  we  have  the  cover,  J  the  cover  and  12 
pounds  equal  to  18  pounds,  or  f  of  the  cover  weighing  6 
pounds;  ^  of  the  cover,  ^  of  6  pounds,  or  2  pounds;  and  |, 
times  2  pounds,  or  4  pounds ;  and  18  pounds  less  4  pounds,  i 
14  pounds,  the  weight  of  the  jar.     Therefore,  etc. 

15  Had  the  vest  cost  ^3  less,  both  had  cost  but  $16,  of 
which  the  coat  cost  3  parts,  the  vest  1,  both  4  parts ;  }  of  ^16 
18  ^4  ;  3  times  $4  is  $12,  the  cost  of  the  coat ;  and  $4  plus  $3, 
or  $7  is  the  cost  of  the  vest.     Therefore,  etc. 

17  Since  |  of  George^s  equaled  J  of  Abel's,  2  halves  would 
(160,  161) 


182  MISCELLANEOUS   EXAMPLES  IN   THE 

equal  twice  f ,  or  J  ;  then  Abel  had  4  parts,  George  6  parta^ 
and  both  10  parts  ;  j\  of  50  cents  is  5  cents ;  6  times  5  cents 
=  30  cents,  George's  money  ;  and  4  times  5  cents  =20  cents, 
Abel's  money.     Therefore,  etc, 

18.  Since  f  equaled  4,  i  would  equal  J  of  4,  orf,  and  f  5 
times  f  or  y  ;  then  the  black  ones  were  7  parts,  the  gray  ones 
10,  and  both  17  ;  j\  of  34  is  2  ;  10  times  2  is  20,  the  num- 
ber of  gray  ones ;  and  7  times  2  is  14,  the  number  of  black 
ones.     Therefore,  etc. 

19.  Since  |  equaled  f,  |  would  equal  |  of  |,  or  i^g,  and  J, 
3  times  /^  or  |f  ;  one  number  is  divided  into  sixteenths,  15  of 
which  equals  the  other,  and  ^  equal  both  ;  Jy  of  62  is  2  ;  16 
times  2  is  32,  the  larger  number;  and  15  times  2  is  30,  the 
smaller  number.     Therefore,  etc. 

20.  Since  J  equals  |,  J  would  equal  J  of  |,  or  -j^,  and  J,  4 
times  ^y  or  y"j  ;  the  value  of  the  contents  is  15  parts,  of  thfl* 
purse  8  parts,  and  of  both  23  paits ;  ^3  of  46  shillings  is  2 
shillings ;  15  times  2  shillings  is  30  shillings,  the  value  of 
the  contents ;  and  8  times  2  shillings  is  16  shillings,  the  value 
of  the  purse. 

22.  Since  from  midnight  to  10  o'clock  is  10  hours,  and  the 
past  time  is  divided  into  3  parts,  the  future  into  2,  and  the 
whole  iuto  5,  we  have  1  part  equal  to  ^  of  10  hours,  or  2  hours ; 
and  3  times  2  hours  is  6  hours,  the  past  time ;  hence  it  was  6 
o'clock. 

23.  Since  |  equals  J,  ^  must  equal  ^  of  J,  or  f,  and  f,  3 
times  f ,  or  f  ;  from  midnight  to  5  o'clock,  p.m.,  is  17  hours, 
and  as  past  time  is  8  parts,  future  9  parts,  and  the  whole  17 
parts,  1  part  equals  1  hour,  and  8  parts  8  hours ;  hence  it  is  8 
o'clock,  A.  M. 

24.  Since  J  equaled  J,  {  would  equal  \  of  J,  or  J,  and  },  4 
times  I,  or  |.  John's  age  was  divided  into  fifths,  4  of  which 
equaled  Peter's,  and  both  equaled  f  of  John's ;  J  of  36  years 
=  4  years,  J  of  John's;  5  times  4  years  =20  years,  John'i 
age;  and  4  timcf  4  years  =16  years,  Peter's  age, 

(161,  162) 


PROGRESSIVE  INTELLECTUAL  ARITHMETIC.         183 

25.  Since  J  equaled  J^*,  f  would  equal  f  ;  and  we  have  what 
:ras  wanting  divided  into  5  parts,  what  was  in  the  bin  into  6 
parts  of  the  same  size,  and  the  whole  capacity  of  the  bin  into 
11  parts;  ^j  of  44  bushels=4  bushels,  1  part;  and  5  times 
4  bushels  =  20  bushels,  what  was  wanting  to  fill  the  bin. 

26.  Since  f  of  what  it  exceeded  equaled  V  of  what  it  lacked. 
I  would  equal  y- ;  and  we  have  what  it  lacked  divided  into  7 
parts,  what  it  exceeded  into  15,  or  the  whole,  83  miles— 39 
miles=44  miles,  divided  into  22  parts ;  ^V  ^^  ^^  ™^^^^  ^^  ^ 
oiles;  7  times  2  miles  is  14  miles,  the  distance  it  lacked  of 
being  S?  miles  ;  and  83  miles  less  14  miles  is  69  miles,  the  dis- 
tance to  Cincinnati. 

27.  Since  f  of  what  it  lacks  of  being  150  miles  equals 
what  it  exceeds  100  miles,  we  have,  the  excess,  3  parts  plus 
the  deficiency,  2  parts,  or  5  parts  in  all,  equal  to  150  miles  less 
100  miles,  or  50  miles ;  }  of  50  miles  is  10  miles ;  3  times  10 
miles  is  30  miles  ;  and  100  miles  plus  30  miles =130  miles,  the 
distance  from  Charleston  to  Columbia. 

105. 

2.  Since  f  equal  4  +  9,  J  will  equal  ^  of  ^  4-  9,  which  is  f 
4-8,  I,  and  4  times  f  +  3,  which  is  ^  -h  12  ;  hence,  the  mother's 
age  is  divided  into  7  parts,  and  8  of  the  same  size +  12  years 
equals  the  father's  age,  or  15  parts  +  12  years  equals  72  years ; 
72  years  less  12  years  is  60  years,  j\  of  60  years  is  4  years, 
and  7  times  4  years  equals  28  years,  the  mother's  age. 

3.  Since  |  equal  f  less  4  rods,  -J-  will  equal  ^  of  J  less  4 
rods,  which  is  |  less  2  rods,  and  §,  3  times  |  less  2  rods,  which 
is  I  less  6  rods ;  hence  what  one  built  equals  6  rods  less  than  ) 
of  what  the  other  built,  and  both  built  -y  of  the  amount  the 
eeccud  did,  less  6  rods ;  38  rods  plus  6  rods,  or  44  rods  equals 
V  ?  fV  ^^  ^4  r^^s  ^^  ^  Todsy  is  A  of  5  times  4  rods  or  20  rode, 
what  the  second  built;  and  38  rods  less  20  rods,  or  18  rods 
fquftis  what  the  first  built. 

4.  Since  \  was  4  more  than  J^  |  would  be  1  more  than  J, 

(162,  163) 


184  MISCELLANEOUS  EXAMPLES  IN   THE 

and  -J,  7  more  than  J  ;  hence  what  Richard  sheared  are  divided 
into  5  parts,-  Hiram's  into  Y  parts  plus  7  sheep,  and  both  into 
12  parts  plus  7  ;  67  less  7  is  60 ;  /_  of  60  =  25,  the  number 
Richard  sheared  ;  67  less  25=42,  the  number  Iliram  sheared. 

5.  Since  |  of  future  time  equaled  |  of  the  past  4-  f  f 
hours,  j-  would  equal  J  of  |  + 1|  hours,  which  is  ^  -f  j\,  and  |, 

5  times  ^4-tj  ^ours,  which  is  -3  +f  hours;  hence  the  future 
time  equals  f  hours  more  than  f  of  the  past,  and  both  past 
and  future  time  equa.  f  of  the  past-hf  hours,  or  24  hours; 
24  hours  less  |-  hours  is  21^  hours,  and  |  of  21^  hours  is  8 
hours,  or  the  past  time;  hence  it  was  8  o'clock  a.  m. 

6.  Since  |  of  what  his  age  lacked  of  being  100  years  equaled 
f  of  what  it  exceeded  64  years,  +  9  years,  |  of  his  age  would 
equal  i  of  J  +  9  years,  whicb  is  J--|- 1  year,  and  f ,  8  times  }  +  l 
year,  which  is  |  -f  8  years ;  hence,  what  his  age  lacked  of 
being  100  years  equaled  8  years  more  than  |  of  what  it  ex- 
ceeded 64  years,  and  \^  of  what  it  exceeded  64,  is  8  years 
less  than  the  difference  between  100  years  and  64  years,  or  36 
years;  36  less  8  is  28  years,  j\  of  28  years  is  2  years,  and 

6  times  2  or  12  years,  is  what  his  age  exceeded  64  years. 

8.  Since  the  body  is  as  long  as  the  head  and  tail,  it  must 
be  I  of  the  length  of  the  fish ;  the  tail  being  as  long  as  the 
head  and  ^  the  body,  must  be  ^  of  the  length  of  the  fish  plus 

7  inches,  and  the  7  inches  it  exceeds  the  {  with  the  7  inches  of 
the  head,  must  equal  the  other  ^  ;  14  is  J  of  4  times  14  or 
66.     Therefore,  etc. 

9.  The  first  price  plus  the  second,  equal  to  J  4-  3  pounds, 
equals  the  third  price;  2  times  j+3,  equal  to  |  of  it +  6,  equals 
the  whole  of  it,  and  6  pounds  must  be  f  of  it ;  |  of  6  pounds  01 
2  pounds  is  }  of  it,  and  5  times  2  =  10  pounds  is  the  whole  of  it. 

Ji^^   Or  it  may  be  solved  like  the  following, 

10.  Since  the  third  dug  as  many  as  the  other  two,  he  dug  ^  , 
and  as  the  first  two  dug  \  less  2  bushels +  5  bushels,  or  3 
bushels  more  than  |^,  those  3  bushels  must  equal  the  difference 

(163,  164) 


PROGRESSIVE    INTELLECTUAL   ARITHMETIC.  185 

Detween  |-  and  ^  of  them,  or  }  of  tliem ;  and  3  bushels  is  J  oi 
6  times  3  bushels  which  is  18  bushels. 

11.  Since  the  distance  from  Avon  to  Bath  is  12  miles  more 
than  the  sum  of  the  other  two  distances  mentioned,  we  have 
the  whole  distance  equal  to  f  of  itself+ 60  miles ;  hence  | 
»f  60  miles  or  20  miles  is  \  of  the  distance  ;  and  5  times  20, 
OT  100  miles  is  the  whole  distance  from  Batavia  to  Corning. 

12.  Since  he  took  $24  more  than  i  of  the  whole  for  shee]» 
and  swine,  and  $7  less  than  f  as  much  for  cattle,  he  took  for 
the  cattle  |2  more  than  \  of  the  whole ;  and  we  have  $18  4-3 
of  the  whole,  +$6,  +1  of  the  whole +  $2,  or  ^^  of  the 
whole +  $26,  equal  to  the  whole  amount;  hence  -^^  of  $26, 
3r  $2  is  2V,  and  24  times  $2,  or  $48  is  what  he  took  for  all. 

13.  Of  ^  that  number  t)f  which  |,  of  J  and  ^  of  ^  of  12  is 
1.  1  and  i  of  ^  of  12  is  6,  ^  of  6  is  2,  and  2  is  1  of  8  times  2, 
or  16.     Therefore,  etc. 

14.  Since  he  earned  |  as  much  as  he  had  spent,  he  only 
lacks  I  of  |=J  of  the  whole,  of  having  as  mucn  as  he  had  at 
first ;  $16.50  is  ^  of  6  times  $16.50,  or  $99.     Therefore,  etc. 

15.  Since  |  equal  f,  |  will  cost  i  of  |  of  an  eagle,  or  $2. 

16.  Since  C  is  |  as  old  as  A,  he  is  4  years  more  than  ^  as 
old  as  B ;  and  as  B's  age  equals  the  sum  of  A's  and  C's,  we 
have  i  of  it  plus  6  years,  plus  ^  of  it  plus  4  years,  or  |  of  it 
+  10  years,  equal  to  itself;  hence  10  years  must  be  }  of  B^s 
age,  and  6  times  10  years  is  60  years  B's  age ;  ^  of  60  is  30, 
30  +  6  is  36,  A's  age ;  and  |  of  36,  or  24  is  C's  age. 

Il,  Since  C  owns  ^  as  much  as  A,  he  owns  6  acres  more 
than  3  as  much  as  B;  and  we  have  what  A  owns,  12  more 
than  J  as  many  acres  as  B,  +  what  C  owns,  6  more  than  f  at 
many  acres  as  B,  equal  to  18  more  than  f  as  many  acres  as  B 
owns,  or  24  acres  more  than  his  farm  ;  hence  6  acres  equals 
J-  of  B's  farm,  8  times  6  is  48  acres,  B's ;  f  of  48  is  36  acres, 
and  36  +  12  equals  48  acres  or  A's  ;  and  ^  of  48,  or  24  acres 
equals  C's. 

(164J 


186  MISCELLANEOUS   EXAMPLES   IN   THE 

106. 

2.  Since  the  son's  age  is  |  of  the  father's,  the  22  years  tho 
father's  age  exceeds  the  son's  must  be  ^  of  the  father's  age ; 
3  times  22  years  is  66  years,  the  father's  age,  and  66  years 
less  22  years  equals  44  years,  the  son's  age ;  at  the  son's  birth 
the  father  was  22  years  old,  in  22  years  from  that  time  each 
would  be  22  years  older,  and  the  son  being  22,  and  the 
father  44  years  of  age,  would  answer  the  condition  of  the  ques 
tion,  and  as  the  son  is  44  now,  44  years  less  22  years,  or  22 
years  since,  he  was  |  as  old  as  his  father. 

3.  At  Helen's  birth  her  sister  was  22  less  9,  or  13  years 
of  age,  and  in  13  years  from  that  time  Helen  would  be  13 
and  her  sister  2  times  13,  or  26  years  of  age;  and  as  Helen 
has  advanced  through  9  of  13  years,  she  has  13  less  9  or  4 
years  more  to  advance. 

Or,  for  brevity,  2  times  9  is  18 ;  22  less  18  is  4.  Therefore,  etc, 

5.  Since  he  took  as  many  from  one  field  and  put  in  the  other 
as  were  there,  and  now  both  have  twice  as  many  as  were  there 
at  first,  the  60  sheep  must  have  been  three  times  ths  number 
before  removing ;  ^  of  60  is  20,  the  number  in  the  smaller ; 
and  20  plus  60,  the  number  in  the  larger  flock,  equals  80,  the 
whole  number. 

6.  Since  both  bins  now  contain  the  same  quantity,  and  each 
2  bushels  more  than  twice  what  was  in  the  less  at  first,  the 
larger  must  have  had  4  bushels  more  than  3  times  the  less ; 
62  less  4  is  48  bushels,  ^  of  48  is  16  bushels,  what  was  in  the 
less;  and  16  bushels +  52  bushels,  what  was  in  the  larger, 
equals  68  bushels.     Therefore,  etc. 

7.  J  less  I  is  2*;j-,  which  by  the  condition  of  the  question,  is 
6  more  than  |  of  his  age;  ^*y  less  |  is  j'^,  6  is  y'j  of  72. 
Therefore,  etc. 

8.  Since  he  received  {  of  his  wages  for  his  summer's  la- 
bor, \  as  much,  or  }  of  them  in  fall  -f  S20,  and  $20  in  springs 
$20 -f  $20=^40  must  be  f  or  i  of  his  wages,  and  3  time* 
|40=$120,  must  be  the  whole  amount. 

(165) 


PROGRESSIVE    INTELLECTUAL   ARITHMETIC.  187 

10.  Since  A  sold  B  J  as  much  as  B  had,  B  now  has  J  o) 
what  he  had  i*t  first,  which  is  J  of  what  A  has  left ;  ^  of  | 
=y''^  is  I ;  and  4  times  j\,  or  |  what  A  has  now  — f  f,  what  A 
has  left ;  f  |  plus  the  J  sold  B,  gives  A  f }  of  B's  before  the 
Bale;  -^\  of  74  is  2,  and  12  times  2  is  24,  the  number  of  acres 
B  had  before  the  sale ;  and  24  acres  plus  f  of  24  acres  equals 
42  acres,  what  he  now  has;  and  74  less  18  acres  leaves  5Q 
acres,  what  A  has  left. 

11.  Since  |  equal  i,  i  will  equal  ^  pf  J  or  J,  and  f,  3  times 
J  or  I ;  hence  as  the  turkeys  equal  J  of  the  chickens,  10  must 
be  the  remaining  \  ;  4  times  10  is  40.     Therefore,  etc. 

12.  Since  |  of  the  price  of  the  coat  equaled  J  of  the  price 
of  the  suit,  ^  would  equal  ^  of  j-  or  J-,  and  f ,  3  times  |  or  J  of 
the  suit ;  and  since  the  coat  cost  f  of  the  price  of  the  suit, 
$15  must  be  f  of  it ;  ^  of  $15,  or  $3  is  |,  and  f  are  8  times 
$3  or  $24.     Therefore,  etc. 

13.  Since  f  times  his  brother's  equaled  J  of  his,  ^  would 
equal  |  of  J  or  y^^  of  his ;  f ,  which  is  3  times  jV,  or  j\  and 
the  14  more  must  be  the  remaining  ^^^  |  of  14  is  2  ;  10 
times  2  or20  equals  what  Daniel  caught,  and  20  less  14  equals 
6,  what  his  brother  caught. 

14.  Since  by  the  conditions  of  the  question  we  have  ^  of 
3^  times  a  number  +15,  equal  to  once  the  number  +15,  or, 
to  avoid  fractions,  2  times  a  number  +30,  equal  to  3^  times 
he  same  number  +  15,  2  times  the  number  is  equal  to  2  times 
the  same  number,  leaving  30  equal  IJ  times  the  number  +  15, 
or  15  equal  to  J  of  the  number;  ^  of  15  or  5  is  i  of  it;  2 
times  5,  or  10  is  the  less  number;  and  3^  times  10,  or  35  is 
the  larger. 

15.  Since  f  equal  /j,  |  will  equal  i  of  j%,  or  /^  ;  and  },  5 
♦.irnes  /^  or  | ;  and  as  the  buggy  cost  |  as  much  as  the  horse, 
the  difference,  $40,  must  be  i  of  the  cost  of  the  horse  ;  3  times 
$40  or  $120  is  the  value  of  the  horse ;  and  |  of  it,  or  $80,  is 
the  value  of  the  buggy. 

16.  5  years  since  the  mother's  ago  was  5  times  Alice's,  ana 

(166) 


188  MISCELLANEOUS   EXAMPLES   IN   THE 

by  the  first  condition  we  have  5  times  Alice's  age  +5  (the 
mother's  age)  equal  to  3  times  Alice's  age  +15  ;  and  since 
3  times  Alice's  age  equals  3  times  her  age,  we  have  2  times 
her  age  +  5  equal  to  15,  or  2  times  her  age  equal  to  10  ;  ^ 
of  10  is  5,  her  age;  5  years  since +  5  equals  10,  her  age  now  ; 
and  3  times  10  or  30  is  her  mother's  age ;  2  times  10  is  20, 
and  30  less  20  is  10,  the  number  of  years  in  which  she  will  be 
I  as  old  as  her  mother.     See  Ex,  3,  in  this  lesson, 

17.  Since  Hobart  has  but  |  of  his  left,  he  lost  \  of  them 
to  Dwight,  which,  by  the  condition  of  the  question,  was  equal 
to  ^  of  D wight's ;  %  must  have  equaled  all  of  D wight's,  and 
the  20  Hobart's  exceeded  Dwight's  must  have  been  \  of  Ho- 
bart's;  3  times  20  =  60  marbles  Hobart  had  ;  and  2  times  20 
=  40  marbles  Dwight  had. 

18.  Since  the  difference  between  the  numbers  is  16,  if  4  be 
taken  from  the  larger  difference  will  be  12,  then  added  to  the 
less  it  will  be  but  8  ;  2 J  times  this  difference,  or  19,  is  equal  to 
3J-  times  less  2f  times  =  if  times  the  less  number;  y^  of  19 
or  1  is  -^^  ;  24  times  1=24,  the  less  number;  and  24-fl6  = 
40,  the  larger  number. 

19.  Since  he  paid  twice  as  much  for  the  rifle  as  for  the 
watch,  and  the  watch  cost  $20,  the  rifle  cost-  2  times  $20,  or  $40. 

20.  Since  C's  age  at  A's  birth  was  5^  times  B's,  and  is  now 
equal  to  the  sum  of  A's  and  B's  ages ;  and  as  the  increase  of 
C's  age  w^ould  just  equal  A's  age,  and  B's  increase  being  the 
same,  the  increase  must  have  been  what  B's  age  lacked  of  being 
equal  to  C's  at  first,  or  4^  times  B's  age  then ;  hence  we  have 

A's  age  now  equal  to  41  times  B's  age  at  first ; 

J^'g     «        *4  a        u     gx      **         "        "      "      " 

Q)g     u        u  u        <'  10  ^^         ^^         ^^       ^^       ^^ 

^ow  if  4  years  be  added  to  B's  age,  |  of  the  sum,  or  4}  times 
/i's  age  as  first +  3  years,  is  equal  to  A's  age,  and  4^  times  B's 
it  first,  which  gives  the  3  years,  is  equal  to  f  of  B's  age  at  first; 
hence  B  was  8  years  old  then,  and  is  now  5^  times  8,  or  44 
vears  old ;  A  is  4|  times  8,  or  36  years  old  ;  and  C  is  10  times 
8,  or  80  years  old. 

(166,   167) 


PROaRESSIVB   INTELLECTUAL  ARITHMETIC.         189 

107. 

2.  In  as  many  hours  as  2  miles,  the  number  he  gained  in  I 
hour,  is  contained  times  2  times  5  miles,  the  distance  to  be 
gained  ;  2  times  5  is  10  miles,  and  2  is  in  10  5  times.  There- 
fore, etc. 

3.  As  many  times  9  rods  as  2,  the  number  of  rods  he  gaina 
in  running  9,  is  contained  times  in  28,  the  number  to  be 
gained;  2  is  in  28  14  times,  and  14  times  9  is  126  rods. 
Therefore,  etc. 

4.  John  will  have  as  many  times  $7  as  $2,  what  he  gains 
on  $7,  is  contained  times  in  $30,  the  whole  gain ;  $2  is  in  $30 
15  times,  and  15  times  $7  is  $105,  what  John  has  saved ;  and 
$105  less  $30  is  $75,  what  Henry  has  saved. 

5.  Since  the  distance  B  ran  is  divid'^d  into  eighths,  1  of 
which  equaled  the  distance  he  was  ahead  of  A,  A  must  have 
run  J  as  far  as  B ;  |  of  84  =  12  rods  is  \  of  the  distance  B  ran, 
and  8  times  12  =  96  rods,  is  B's  distance 

6.  Since  $25  is  \  of  what  B  and  C  paid,  they  paid  4  times 
$25  or  $100,  which,  with  the  $25  A  paid,  makes  $125,  th© 
cost  of  the  horse ;  and  since  B  paid  |  as  much  as  A  and  C, 
they  paid  3  parts  and  he  2  parts ;  that  is,  5  parts  equal  the 
whole  cost ;  f  of  $125  is  $50,  what  B  paid ;  and  $50  plus 
the  $25  A  paid  equals  $75  which,  taken  from  $125,  leaves  $50, 
what  C  paid. 

8.  Since  the  minute  hand  passes  over  12  spaces  while  the 
hour  hand  passes  over  1,  tbe  minute  hand  gains  11  spaces  on 
the  hour  hand  for  every  12  spaces  it  passes  over,  and  it  would 
pass  as  many  times  12  spaces  as  11,  the  number  it  gains  in 
passing  12,  is  contained  times  in  45  spaces,  the  number  to  be 
gp^ned  after  3  o'clock  before  they  are  opposite ;  11  is  in  45  4  p"| 
timus,  and  4Jy  times  12  is  49 jV  spaces.  Therefore  it  would 
be  49y*y  minutes  past  3  o'clock. 

9.  Since  3  of  the  hound's  leaps  equal  6  of  the  fox's,  1  will 
equal  I  of  6,  or  2  of  the  fox's,  and  4,  4  times  2,  or  8  of  the 
fox's ;  hence  the  fox  will  take  as  many  times  7  leaps  as  1,  the 

9.R.P.  (167,  168) 


190  MISCELLANEOUS   EXAMPLES,   ETC. 

number  the  hound  gains  on  the  fox  in  making  7,  is  contained 
times  in  40,  the  number  of  leaps  to  be  gained ;  1  is  in  40,  40 
times,  and  40  times  1  is  280.     Therefore,  etc. 

10.  Since  the  distance  the  sheep  ran  was  divided  into  5  parts, 
8  of  which  equaled  the  distance  between  them,  the  whole  dis- 
tance equaled  8  parts ;  |  of  80  rods  or  10  rods  is  1  part,  and 
3  times  10  or  30  rods  eq^ials  3  parts,  or  the  distance  betwee 
thorn. 

11.  Since  the  interest  at  5  per  cent.,  for  2  years  7  months 
and  6  days,  is  y'^^  of  the  principal,  the  amount  will  be  m  ; 
j{^  of  $2260  is  $20,  and  100  times  $20r=$2000,  the  sum  at 
interest ;  and  since  B's  money  equaled  f  of  A's,  the  whole 
equaled  |  o^  A's ;  }  of  $2000  is  $400  or  i  of  what  A  had  in, 
which  is  I  of  all,  and  5  times  $400  or  $2000  equals  A's ; 
$400  equaled  \  of  what  B  had  in,  which  is  |  of  all,  and 
8  times  $400  or  $3200  equals  B's. 

12.  Sinc(>  B's  fortune  is  1^  times  A's,  \  of  A's  is  equal  to  ^ 
of  Ws,  and  Ih*^  mterest  on  it  for  5  years  at  6  per  cent,  would 
equal  j\  of  it  ^  of  $600,  or  $200,  is  yV  of  10  times  $200, 
or  $2000;  i  ot  $2000,  or  $1000*,  equals  what  each  had  in  ;  2 
times  $1000,  or  $2000,  equals  A's  fortune ;  and  3  times  $1000, 
or  $3000,  equals  B'c. 

13.  Since  he  lost  8  per  cent,  or  -fj  of  the  cost  on  the  sale, 
he  sold  for  ||  of  the  ca^ ;  hence  \  of  his  calves  and  |  of  his 
shf^ep  cost  $25,  and  4  times  $25,  or  $100  is  the  cost  of  all 
the  calves  and  |  of  the  sheep ;  this  exceeds  the  whole  cost  by 
$24,  which  must  equal  the  cost  of  the  |  of  the  sheep  ovei 
the  whole  number ;  -J-  of  $24  is  $C,  and  5  times  $8  is  $40 
which  would  buy  20  sheep  at  $2  ;  ^76  lesj.  $40  givep  $36 
for  calves,  which  would  buy  12  calves  at  $3.     Therefore,  etf 

(168) 


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